Gravitational radiation from gamma-ray bursts as observational opportunities for LIGO and V
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Gravitational radiation from gamma-ray bursts as observational opportunities for LIGO and VIRGO Maurice H.P.M.van Putten 1,Amir Levinson 2,Hyun Kyu Lee 3,Tania Regimbau 1,Michele Punturo 4,and Gregory M.Harry 1ABSTRACT Gamma-ray bursts are believed to originate in core-collapse of massive stars.This produces an active nucleus containing a rapidly rotating Kerr black hole of mass M H and angular velocity ?H ?1/2M H ,surrounded by a uniformly mag-netized torus of angular velocity ?T =η?H represented by two counter-oriented current rings.We quantify black hole-spin interactions with the torus and charged particles along open magnetic ?ux-tubes subtended by the event horizon at a ?-nite half-opening angle θH .A major output of E gw ?4×1053(η/0.1)(M H /7M ⊙)erg is radiated in gravitational waves of frequency f gw ?500(η/0.1)(7M ⊙/M H )Hz by a quadrupole mass-moment in the torus when its minor-to-major ra-dius is less than 0.3260.The durations correspond to the lifetime T s of black hole-spin,determined by a stability condition of poloidal magnetic ?eld energy-to-kinetic energy <1/15in the torus.Consistent with GRB-SNe,we ?nd (i)T s ?90s (tens of s,Kouveliotou et al.1993),(ii)aspherical SNe of kinetic en-ergy E SN ?2×1051erg (2×1051erg in SN1998bw,H¨o ?ich et al.1999)and (iii)GRB-energies E γ?2×1050erg (3×1050erg in Frail et al.2001),upon associating θH with poloidal curvature of the magnetosphere.GRB-SNe occur perhaps about once a year within D =100Mpc.Correlating LIGO/Virgo detec-tors enables searches for nearby events and their spectral closure density 6×10?9
around 250Hz in the stochastic background radiation in gravitational waves.At current sensitivity,LIGO-Hanford may place an upper bound around 150M ⊙in GRB030329.Upcoming all-sky supernovae surveys may provide distances to GRB-SNe,conceivably coincident with weak wide-angle GRB emissions similar to the nearby event GRB980425/SN1998bw.Detection of E gw thus provides a method for identifying Kerr black holes by calorimetry.
1.Introduction
Gravitational wave detectors LIGO(Abramovici et al.1992)and Virgo(Bradaschia et al.1992),are broad-band detectors,most sensitive in20-1000Hz.They introduce new opportunities for probing strongly-gravitating astrophysical sources and the stochastic back-ground radiation in gravitational waves–see(Cutler&Thorne2002)for a recent overview. Notable candidates for burst sources of gravitational radiation are binary coalescence of neu-tron stars and black holes(Narayan,Piran&Shemi1991;Phinney1991),newborn neutron stars(Ferrari et al.2003),and gamma-ray bursts:long bursts associated with supernovae (van Putten2001;M′e sz′a ros2002;van Putten&Levinson2003)and short bursts,conceiv-ably associated with black hole-neutron star coalescence.Collectively,these astrophysical sources may contribute appreciably to the stochastic background in gravitational waves.Of particular interest is the quest for identifying Kerr black holes in the Universe,and these objects may be at the center of cosmological GRBs.
The events GRB980425/SN1998bw(Galama et al.1998)and GRB030329(Hjorth et al.2003;Stanek et al.2003)demonstrate that long GRBs are associated with Type Ic supernovae.This provides considerable support for GRBs as core-collapse events in massive stars in binaries(Woosley1993;Paczynski1998;Brown et al.2000),and hence an association with star-forming regions.A correlation of the current?ux-limited sample of33GRBs with individually measured redshifts with the cosmic star formation rate shows a true-to-observed GRB event rate of450(van Putten&Regimbau2003),which is very similar to the beaming factor of500obtained from a subsample of GRBs with achromatic breaks in their light curves (Frail et al.2001).The true GRB-event rate is hereby about one per year within a distance of100Mpc.The associated supernova and late-time GRB-afterglows may present wide-angle optical and radio emissions as orphan transients to nearby events.Because the event rate of GRB980425/SN1998bw is roughly consistent with the true GRB-event rate,nearby events are conceivably detectable by extremely weak but non-vanishing GRB emissions at large viewing angles.
Here,we report on signal-to-noise ratios for gravitational radiation from GRBs from rotating black holes in matched?ltering and in correlating two detectors of LIGO and Virgo, both in targeting GRBs as nearby point sources and in their contribution to the stochastic background radiation.Detection of these emissions from GRBs with measured redshifts enables calorimetry on their inner engines,as a method for rigorously identifying Kerr black holes as objects in the Universe.
We propose a radically new model(Fig.1):GRB-SNe produced by an active nucleus in a remnant stellar envelope comprising a Kerr black hole,surrounded by a uniformly magnetized torus in suspended accretion represented by two counter-oriented current rings
(van Putten1999;van Putten&Ostriker2001;van Putten2001;van Putten&Levinson 2002,2003).Our model predicts an energetic output in gravitational waves from black hole-spin energy with energy output and frequency
E gw?0.2M⊙ η7M⊙ ,f gw?500Hz ηM H (1) emitted by a non-axisymmetric torus surrounding the black hole.Here,we measure energy in units of M⊙=2×1054erg,andηdenotes the ratio of the angular velocity of the torus to that of the black hole of mass M H,which represents the e?ciency of converting black hole-spin energy into radiation by the torus.Because the system is relativistically compact,most of the torus output is in gravitational radiation by multipole mass-moments(van Putten2001). It has not eluded us,that the output in gravitational radiation(1)surpasses Eγ?3×1050 erg in gamma-rays(Frail et al.2001)by three orders of magnitude.
A quadrupole mass-moment associated with a mass-inhomogeneityδM T produces a luminosity(Peters&Mathews1963)
L gw=32
5
(M H/R)5(δM T/M H)2,(2)
whereω?M1/2/R3/2denotes the orbital frequency of the torus with major radius R, M=(δM T M H)3/5/(δM T+M H)1/5?M H(δM T/M H)5/3denotes the chirp mass,and F(e) denotes a geometric factor representing the ellipticity e of the oribital motion.Application of (2)to PSR1913+16with ellipticity e=0.62(Hulse&Taylor1975)provided the?rst evidence for gravitational radiation consistent with the linearized equations of general relativity to within0.1%(Taylor1994).Here,we apply the right hand-side of(2)to a non-axisymmetric torus around a black hole,whose mass-quadrupole inhomogeneityδM T is determined self-consistently in a state of suspended accretion for the lifetime of rapid spin of the black hole.A quadrupole mass-moment appears spontaneously as a Papaloizou-Pringle wave(Papaloizou &Pringle1984)whenever the torus is su?ciently slender,i.e.,for a ratio b/R<0.3260, where b denotes the minor radius of the torus(van Putten2002).In the suspended accretion state,most of the black hole-spin energy is dissipated in the event horizon for typical ratios η～0.1of the angular velocity of the torus to that of the black hole.Hence,the lifetime of rapid spin of the black hole is e?ectively determined by the rate of dissipation of black hole-spin energy in the event horizon,itself bounded by a?nite ratio E B/E k<1/15of the poloidal magnetic?eld energy-to-kinetic energy in the torus(van Putten&Levinson2003). This gives rise to long durations of tens of seconds for the lifetime of rapid spin of the black hole.The resulting gravitational wave-emissions should be limited in band width,changing in frequency about10%during the emission of the?rst50%of its energy output.This change mirrors a decrease of10%in the angular velocity of a maximally-spinning black hole
in converting50%of its spin-energy.Thus,gravitational radiation is connected to Kerr black holes,representing a connection between the linearized equations of general relativity and,respectively,fundamental objects predicted by the fully nonlinear equations of general relativity.
To date,short GRBs appear to have featureless afterglow emissions,and their cosmolog-ical origin is based on an isotropic distribution in the sky and a
Gravitational radiation associated with collapsars has been considered in a number of other studies(Nakamura&Fukugita1989;M¨o nchmeyer et al.1991;Bonnel&Pringle1994; Davies et al.2002;Fryer et al.2002;Mineshige et al.2002;Kobayashi&Meszaros2002),also in model-independent search strategies associated with GRBs(Finn,Mohanty&Romano 1999;Modestino&Moleti2002).These studies focus on gravitational radiation produced by the release of gravitational binding energy during collapse and in accretion processes on a newly formed black hole(e.g,Fryer et al.(2001)).We note that accretion?ows are believed to be strongly turbulent,which may imply a broad spectrum of gravitational radiation. Forementioned studies on gravitational radiation in core-collapse of a massive star do not invoke the spin-energy of a newly formed black hole.They appear to indicate an energy output,which leaves a range of detectability by current ground based detectors of up to about10Mpc.These events should therefore be considered in the context of core-collapse events independent of the GRB phenomenon,in light of current estimates on the local GRB event rate as referred to above.Currently published bounds on gravitational wave emissions from GRBs are provided by bar detectors(Tricarico et al.2001;Astone et al.2002).These
studies and results are important in identifying various detection strategies and channels for producing gravitational waves.We suggest that the design of optimal detection strategies for gravitational radiation from GRBs may be facilitated by a priori knowledge from a speci?c model.
In the presented studies,we describe a model for GRB-SNe from rotating black holes which is consistent with the observed durations and true energies in gamma-rays from mag-netized baryon-poor jets subtended by the event horizon of a black hole(1),the observed total kinetic energies in an associated supernova,possibly radio loud,with aspherical distri-bution of high velocity ejecta(2),and X-ray line-emissions produced by underlying contin-uum emissions(3).On this basis,we predict band-limited gravitational wave line-emissions contemporaneous with the GRB according to the scaling relations(1)at an event rate of probably once a year within a distance of100Mpc.
In§2,we describe elements of current GRB-phenomenology.In§3-4we summarize a theory of GRB-supernovae from rotating black holes.In§5,we discuss line-broadening in response to Lense-Thirring precession.A semi-analytical estimate is given of the con-tribution to the stochastic background in gravitational waves in§6.In§7,we present the dimensionless characteristic strain-amplitudes,and in§8we calculate the signal-to-noise ra-tios in advanced LIGO and Virgo operations in various detection strategies.§8introduces a proposed detection strategy for time-frequency trajectories of slowly varying line-emissions. We summarize our?ndings in§10.
2.Phenomenology of GRB-supernovae
X-ray localization of GRBs by BeppoSax introduced the post-BATSE development of providing a sample of GRBs with individually measured redshifts(Table1).The recent HETE-II burst GRB030329has greatly enhanced our con?dence in a GRB association to Type Ib/c supernovae,based on a similarity of its optical light curve and emission-lines to that of SN1998bw(Hjorth et al.2003;Stanek et al.2003).This observational association supports a GRB event rate which is locked to the star-formation rate,as core-collapse events in evolved massive stars(Woosley1993;Paczynski1998;Brown et al.2000). This is consistent with recent statistical correlations over a boad range of redshifts(Scheafer et al.2001).It appears that Type Ib/c SNe occur only in spiral galaxies(Cappellaro et al. 1997).Further evidence for the GRB-supernova association is found in X-ray line-emissions in GRB970508(Piro et al.1999),GRB970828(Yoshida et al.1999),GRB991216(Piro et al. 2000),GRB000214(Antonelli et al.2000)and GRB011211(Reeves et al.2001).
When attributing the X-ray line-emissions to excitation by high-energy continuum emis-sions with energy E r,Ghisellini et al.(2002),based on Lazzati et al.(2002),estimate substantial lower bounds for E r:long-lived iron-line emissions in GRB99126may require E r≥4×1052erg,whilst lines from lighter elements in GRB011211may likewise require E r≥4.4×1051.These lower bounds point towards an energy reservoir in excess of that
required for the true GRB-energies Eγ.We interpret this as support for the notion that GRB inner engines may be processing other channels,which are contemporaneous with the baryon-poor input to the GRB-afterglow emissions.
The discovery of achromatic breaks in the light curves of some of these GRBs allowed the determination of the true GRB-energies of about3×1050erg,upon correcting the isotropic equivalent emissions by an observed beaming factor of about500(Frail et al.2001).The redshift distribution of the?ux-limited sample of33GRBs,locked to the SFR,allows us to estimate the unseen-to-observed GRB rate to be about450based on a log-normal?t of the peak-luminosity function–see Fig.2(van Putten&Regimbau2003).This result is very similar to the observed beaming factor of500.With the ratio450of the unseen-to-observed GRBs,the true-but-unseen GRB event rate to about0.5×106per year or,equivalently,1 per year within a distance of100Mpc.This event rate agrees with that based on a fraction of about1%of SNe Ib/c that might be associated with GRBs(Salmonson2001).Note that our event rate is a lower bound on the rate of formation of GRB inner engines,since we are seeing only those events in which the remnant stellar envelope is successfully penetrated by a baryon poor jet,e.g.,when the jet is su?ciently collimated.The true rate of formation of GRB inner engines a therefore an important open question.
The relatively narrow distribution of GRB-energies around3×1050erg is indicative of a standard energy reservoir(Frail et al.2001).An anticorrelation between the observed opening angle and redshift points towards wide-angle GRB emissions which are extremely weak,as in GRB980425.Given that the event rate of GRB980425at D=34Mpc is roughly consistent with1per year within D=100Mpc,these wide-angle emissions may also be standard.Thus,we are led to consider strongly anisotropic GRB emissions in response to out?ow in two directions along the rotational axis of the progenitor star,accompanied by extremely weak GRB emissions in all directions(see further Eichler&Levinson(1999); Zhang&Meszaros(2002);Rossi et al.(2002)).In this regard,GRB980425(Eγ,iso?1048erg, z=0.0085)is not anomalous,and GRB030329(Eγ?3×1049erg,z=0.167)is intermediate (Price et al.2003).GRBs may be geometrically standard,in that this anisotropy is similar in its angular distribution in all sources.In this event,the inferred beaming factor depends on redshift,i.e.,is a function of the?ux-limit in the sample at hand.Current GRB samples with individually measured redshifts,including that of Frail et al.(2001),are dominated by sources with redshifts around unity.
Recent detection of linear polarization GRB021206provides evidence of synchrotron ra-diation in magnetized out?ows,which may indicate large-scale magnetic?elds of or produced by the inner engine(Coburn&Boggs2003).Afterglow emissions to GRB030329include op-tical emissions(Price et al.2003)with intraday deviations from powerlaw behavior(Uemura et al.2003),possibly re?ecting an inhomogeneous circumburst medium or latent activity of the inner engine(Chevalier&Li1999;Price et al.2003).
Furthermore,Type Ib/c SNe tend to be radio-loud(Turatto2003),as in SN1990B(van Dyk et al.1993).This includes GRB980425/SN1998bw(Kulkarni et al.1998;Iwamoto1999) as the brightest Type Ib/c radio SN at a very early stage(Weiler et al.2001).GRB030329 might also feature some radio emission associated with the associated SN2003dh(Willingale et al.2003).Radio emissions in these SNe are well described by optically thick(at early times)and optically thin(at late times)synchrotron radiation of shells expanding into a circumburst medium of stellar winds from the progenitor star(Li&Chevalier1999).All core-collapse SNe are strongly non-spherical(H¨o?ich et al.2001),as in the Type II SN1987A (H¨o?ich1991)and in the Type Ic SN1998bw(H¨o?ich et al.1999),based,in part,on polar-ization measurements and direct observations.Observed is a rotational symmetry with axis ratios of2to3.This generally re?ects the presence of rotation in the progenitor star and/or in the agent driving the explosion.Aforementioned X-ray line-emissions in GRB011211may be excited by high-energy continuum emissions of much larger energies(Ghisellini et al. 2002).For Type Ib/c supernovae association with a GRB,these considerations have led some to suggest the presence of new explosion mechanism(Woosley et al.1999).
Ultimately,GRB-supernova remnants take the form of a black hole in a binary with an optical companion,surrounded by a supernova remnant(van Putten&Levinson2003). This morphology is illustrated by RX J050736-6847.8(Chu et al.2000),if its X-ray binary harbors a black hole.The accreting binary may hereby appear as a soft X-ray transient in the scenario of Brown et al.(2000).
3.GRB-supernovae from rotating black holes
GRBs are believed to be produced in core-collapse of massive stars(Woosley1993), whose angular momentum most likely derives from orbital angular momentum during a common envelope stage(Paczynski1998).The common envelope stage must ensue only when the progenitor star is evolved(Brown et al.2000).Core-collapse in this scenario describes an initial implosion which produces an active nucleus consisting of a Kerr black hole surrounded by a magnetized torus,inside a remnant stellar envelope.Evidently,the black hole-torus system is relativistically compact,in that its linear size is on the order of its
Schwarzschild radius.The nucleus is active,by virtue of the spin-energy of the black hole,
as outlined below.
3.1.MeV nuclei in a remnant envelope
We consider a uniformly magnetized torus around a rapidly rotating black hole in its
lowest energy state.This introduces radiative processes through two spin-interactions with
the black hole:a spin-connection to the torus and a spin-orbit coupling to charged particles
along open magnetic?ux-tubes.The torus hereby catalyzes black hole-spin energy into grav-
itational radiation,accompanied by winds,thermal and MeV-neutrino emissions.The open
magnetic?ux-tubes produce baryon poor out?ows and subsequent high-energy radiation.
This result is a broad range of emissions.In what follows,?H denotes the angular velocity
of the black hole and?T denotes the angular velocity of the torus.
A uniformly magnetized torus introduces an ordered poloidal magnetic?ux,represented
by two counter-oriented current rings.The equilibrium moment of the black hole preserves
essentially uniform and maximal horizon?ux.When viewed in poloidal cross-section,the
inner and the outer torus magnetosphere are topologically equivalent to that of a rapidly
rotating neutron star with angular velocities?(?H??T)and?T,respectively.The interface between the inner and the outer torus magnetospheres is an ellipsoidal separatrix(Fig.3).
An open magnetic?ux-tube subtended by the event horizon of the black hole may form by
moving the separatrix of the inner and outer torus magnetosphere to in?nity(van Putten
&Levinson2003).We emphasize that the equivalence between the inner face of the torus
and a pulsar is exact in topology,yet refers to similar but not identical physical states.For
example:there exist corresponding annuli of B=0between the last closed?eld-lines of the
torus and the event horizon,and between the last closed?eld-lines of the pulsar and in?nity,
where the former features a spark gap which is absent in the latter;both the event horizon
and asymptotic in?nity are null-surfaces,where the former but not the latter is endowed
with?nite surface gravity and the no-hair theorem.
Equivalently to pulsars,the inner face of the torus emits negative angular momentum
Alfv′e n waves into the event horizon,while the outer face emits positive angular momentum
Alfv′e n waves to in?nity.Both emissions satisfy causality.The torus hereby develops a state
of suspended accretion,described by balance of energy and angular momentum?ux received
by the spin-connection to the black hole and emitted in various channels.In response to this
catalytic process,the black hole evolves by conservation of energy and angular momentum
consistent with the no-hair theorem.While most of the spin-energy is dissipated in the event
horizon of the black hole,most of the black hole-luminosity is incident onto the inner face of
the torus.The latter represents substantial fraction of black hole-spin energy,given by the ratio of the angular velocity of the torus to that of the black hole,and is mostly reradiated into gravitational radiation by multipole mass moments in the torus.Dominant emissions in gravitational radiation are typical for systems whose linear size is on the order of their Schwarzshild radius,and similar to those from new born neutron stars(Shapiro&Teukolsky 1983).The catalytic emissions by the torus last for the lifetime of rapid spin of the black hole.Contemporaneously,the torus radiates a minor output in baryon-rich magnetic winds, thermal and MeV-neutrino emissions.The remnant stellar envelope is hereby irradiated from within by high-energy radiation coming o?the torus winds.This associated outgoing radial momentum drives a non-spherical supernova with subsequently X-ray line-emissions when the expanding envelope reaches optical depth of unity or less.Ultimately,this leaves a supernova remnant around a black hole in a binary with an optical companion.
A spin-orbit coupling between the black hole and charged particles creates charged out?ows along open magnetic?ux-tubes subtended by the event horizon of the black hole. Moving the separatrix in Fig.3to in?nity by a stretch-fold-cut creates these open?ux-tubes with a?nite opening angle on the horizon(Fig10in van Putten&Levinson(2003)).In the lowest energy state of the black hole,the spin-orbit coupling introduces the identity (Hawking1976;van Putten2000)
e EMFν=ν?H(in eV)(3) on charged particles o
f angular momentumν=eAφin their Landau states alon
g magnetic ?ux-surfaces Aφ=const.,where2πAφdenotes the magnetic?ux and?e the charge of the electron.This coupling enables the black hole to convert mechanical work by rotation into electrical currents along the axis of rotation.Mediated by frame-dragging,this process is causal and local in origin,in that it corresponds to the line-integral of the electric?eld along the magnetic?eld in Wald(1974).It may be compared wit
h currents induced by Lorentz forces on charged particles when forced to cross magnetic?eld-lines,with the remarkable distinction that the former produces an EMF parallel to and the latter produces an EMF orthogonal to magnetic?eld-lines.The net current along the open?ux-tube is determined by the detailed state of the magnetosphere formed by charge-separation and the boundary conditions on the horizon and at in?nity,as discussed in van Putten&Levinson(2003).In response to the induced charged out?ows,the black hole evolves by conservation of energy, angular momentum and charge consistent with the no-hair theorem and,if present,current closure.The fraction of black hole-spin energy released in baryon-poor out?ows along open magnetic?ux-tubes tends to be small for a?nite horizon half-opening angle on the horizon. The out?ows consist primarily ofγe±,Poynting?ux and kinetic energy in baryonic contam-inants.We associate these emissions with the baryon-poor input to GRBs.These represent dissipation of kinetic energy according to the internal shock model(Rees&M′e sz′a ros1994;
Piran1999;M′e sz′a ros2002).Note that the true energy Eγ?3×1050(Frail et al.2001)in gamma-rays represents a mere0.01%of the rotational energy of a stellar mass black hole.The magnetized baryon-poor out?ows are surrounded by magnetized baryon-rich winds coming o?the torus.The latter may provide collimation to the former(Levinson&Eichler2000). Scattering of photons onto the boundary layer between the two produces highly polarized radiation,which may exceed that attainable in synchrotron emissions within the collimated baryon-poor jet(Eichler&Levinson2003).
Our model is parametrized as follows.The nucleus contains a black hole of mass M H, angular momentum J H=aM H and electric charge q,where a/M H=sinλdenotes the speci?c angular momentum.Because all particles approaching the event horizon assume the angular velocity?H=tan(λ/2)/2M H,there is a magnetic momentμH=qa aligned with its axis of rotation(Carter1968).It preserves essentially uniform and maximal horizon ?ux at arbitrary rotation rates in equilibrium with a surrounding torus magnetosphere of ?eld-strength B withμH?2BM H r2H,where r H=2M H cos2(λ/2)denotes the radius of the event horizon(van Putten2001).Upon balance with various radiation channels,the torus develops a state of suspended accretion at MeV temperatures in equilibrium with its input in energy and angular momentum through the spin-connection to the central black hole. The fractions of black hole-spin energy radiated into various channels depend on the angular velocityηof the torus relative to that of the black hole,the slendernessδ=b/2R of the torus in terms of one-half the ratio of the minor radius b to the major radius R,and the mass-fractionμ=M T/M H of the torus mass M T relative to M H.The half-opening angle of the open magnetic?ux-tube on the event horizon of the black hole is denoted byθH.
In the suspended accretion state,the torus assumes a state of di?erential rotation which exceeds that of Keplerian motion.The inner face is super-Keplerian,while the outer face is sub-Keplerian due to competing surface stresses on the inner face and the outer face of the torus by,respectively,the action of the black hole and and torus winds to in?nity. Both faces may develop surface waves,similar to water waves in channels of?nite depth, since the e?ective gravity is outgoing in the inner face and ingoing on the outer face.In the corotating frame,the inner and outer faces may carry retrograde,respectively prograge waves.This allows the former to decrease its angular momentum and the latter to increase it angular momentum.Any coupling between these the inner and outer surface waves would lead to angular momentum transfer from the inner to the outer face,which may result in instability.This picture describes the Papaloizou-Pringle waves(Papaloizou&Pringle 1984),originally discovered as an azimuthal symmetry breaking instability in tori of in?nite slenderness(b/R→0).An extension of this theory to tori of?nite slenderness(b/R=0?1) shows that the m=0wave-modes become successively unstable as the torus becomes more
slender.We have(van Putten2002)
b/R<0.7506,0.3260,0.2037,0.1473,0.1152,···,0.56/m,(4) for the onset of instability of the m?th buckling mode at the point of Rayleigh stability (stability of the m=0mode between the two faces).In the proposed suspended accretion state,the amplitude of the resulting quadrupole mass-moment,possibly accompanied by higher order mass-moments,saturates in energy and angular momentum balance between input from the black hole and output in forementioned radiation channels.These hydrody-namic instabilities may be accompanied by other instabilities,such as those associated with a strong magnetic?eld.The strength of the poloidal magnetic?eld-energy is subject to the stability criterion(van Putten&Levinson2003)
E B
,(5)
15
based on a linear analysis of non-axisymmetric buckling modes.A similar stability analysis for the tilt mode replaces the right hand-side in(5)with1/12.This upper bound on the magnetic?eld-strength sets a lower bound on the dissipation rate of black hole-spin energy in the event horizon,and hence a lower bound on the lifetime of rapid spin of the black hole. For the parameters at hand,the lifetime of black hole-spin is hereby tens of seconds(below). The torus itself develops MeV temperatures in a state of suspended accretion(van Putten &Levinson2003).
3.2.Radiatively supernovae powered by black hole-spin energy
The remnant stellar envelope is irradiated from within by high-energy continuum emis-sions from powerful torus winds,which were released during the preceding GRB.This con-tinuum emission radiatively drives a supernova by ejection of the remnant envelope and, when the remnant envelope has expanded su?ciently for its optical depth to this continuum emission has dropped below unity,excites X-ray line-emissions as observed in GRB011211 (Reeves et al.2001;van Putten2003).This supernova mechanism is novel in that the supernova-energy derives ab initio from the spin-energy of the black hole,and is otherwise similar but not identical to pulsar driven supernova remnants by vacuum dipole-radiation (Ostriker&Gunn1971),and magnetorotational driven Type II supernovae by Maxwell stresses(Bisnovatyi-Kogan1970;LeBlanc&Wilson1970;Bisnovatyi-Kogan et al.1976; Wheeler et al.2000;Akiyama et al.2003)and associated heating(Kundt1976).
The energy output in torus winds has been determined in a detailed calculation on the suspended accretion state,and is found to be consistent with the lower bound of Ghisellini
et al.(2002)on the energy in continuum emissions for the line-emissions in GRB011211 (van Putten2003).In our proposed mechanism for supernovae with X-ray line-emissions, therefore,we envision e?cient conversion of the energy output in torus winds into high-energy continuum emissions,possibly associated with strong shocks in the remnant envelope and dissipation of magnetic?eld-energy into radiation.We note that the latter is a long-standing problem in the pulsars,blazars and GRBs alike(see Levinson&van Putten(1997) and references therein).Conceivably,this process is aided by magnetoturbulence downstream (Layzer1965;Burbidge1967).These supernovae will be largely non-spherical,as determined by the collimation radius of the magnetic torus winds,see,e.g.,Camenzind(1990)and references therein.
The proposed association of the X-ray line-emissions with the supernova explosion, based on the same underlying large energy in high-energy continuum emissions within the remnant envelope,leads to the prediction that the intensity of line-emissions and the kinetic energy in the ejecta are positively correlated.
3.3.Baryon loading in the magnetized baryon-poor jet
A small fraction of the black hole spin energy is channeled along the black hole rotation axis in the form of baryon-poor out?ows along an open magnetic?ux-tube,as input to the estimated GR
B energies Eγ=3×1050erg of Frail et al.(2001).The baryon content and the loading mechanism of these jets(and essentially of GRB?reballs in any model)is yet an open issue.In one scenario proposed recently(Levinson&Eichler2003)baryon loading is accomplished through pickup of neutrons di?using into the initially baryon-free jet from the hot,baryon-rich matter surrounding it.The free neutrons are produced in the hot torus that maintains temperatures of the order of a few MeV,and stream with the baryon-rich wind emanating from the torus to a radius of～1010cm,above which they recombine with protons to form4He.The pickup process involves a collision avalanche inside the baryon-poor jet (BPJ),owing to the large optical depth for inelastic nuclear collisions contributed by the inwardly di?using neutrons.The hadronic shower saturates quickly,giving rise to a viscous boundary layer at the outer edge of the BPJ where most of the pickups occur.This boundary layer has a moderate bulk Lorentz factor.The Lorentz factor of the BPJ core,where the baryon density is smaller is much larger initially.The picked-up neutrons in the hot boundary layer can remain free up to a radius of about1013cm where they recombine,and continue to di?use into the BPJ core as the BPJ expands.This leads to further collisions in the BPJ core with highly-relativistic baryons coming from below.The total number of picked-up neutrons is estimated to be～1049.5,although it depends somewhat on the out?ow parameters.The
asymptotic bulk Lorentz factor of the BPJ is established in this model at rather large radii (～1012cm)after neutron pickup is completed,and lies in the range between a few hundreds to a few thousands.The expected variation of the Lorentz factor across the BPJ should give rise to orientation e?ects that need to be assessed yet.The inelastic nuclear collisions inside the BPJ lead to e?cient emission of very high-energy neutrinos(energies well above1TeV) with a very hard spectrum.The neutrino?uxes predicted are high enough to be detected by the upcoming km3neutrino detectors,even for a source at a redshift of1.
4.Timescales and radiation energies
Theoretical predictions in the model of GRBs from rotating black holes can be compared with observations on durations and true GRB energies.We shall do so in dimensionless form, relative to the Newtonian timescale of orbiting matter and the rotational energy of a rapidly rotating Kerr black hole of mass7M⊙.
The durations T90are given by the time of activity of the inner engine of the GRB(Piran &Sari1998).We propose to identify the lifetime of the inner engine with that timescale T s of rapid spin of the black hole.This timescale is e?ectively set by the rate of dissipation of black hole-spin energy in the event horizon,by spin-down against the surrounding magnetic ?eld of strength
B c?1016G 7M⊙R 2 M T
7M⊙ η0.03 ?1/2.(7) This estimate is consistent with durations of tens of seconds of long gamma-ray bursts (Kouveliotou et al.1993).This gives rise to the large parameterγ0=T s?T,
γ0=1×105 η0.03 ?1/2(8) consistent with the observed ratio T90?T～105.
The true energy in gamma-rays is attributed to baryon-poor energy out?ow along an open magnetic?ux tube along the axis of rotation of the black hole.As the torus develops
MeV temperatures in the suspended accretion state,it supports a surrounding powerful baryon-rich wind with a mass-loss rate of about1030g s?1(van Putten&Levinson2003). We envision that these torus winds introduce a change in poloidal topology of the inner torus magnetosphere,upon moving the separatrix out to in?nity.This creates an open magnetic ?ux-tube with?nite horizon half-opening angleθH.The open?ux-tube forms an artery for a small fraction of black hole-spin energy,releasing magnetized baryon-poor out?ows.For a canonical value??15%of the e?ciency of conversion of kinetic energy-to-gamma rays(for various estimates,see Kobayashi et al.(1997);Daigne&Mochkovitch(1998);Panaitescu& Kumar(2000);Guetta et al.(2001)),we have,based on van Putten&Levinson(2003),a small parameterγ1=Eγ/E rot,
γ1??θ4H.(9) Here,we propose to attributeθH to poloidal curvature in the inner torus magnetosphere,i.e.,θH?M H/R for a magnetic?eld which is orthogonal to the polar regions of event horizon. This gives Eγ?2×1050(?/0.15)(η/0.1)8/3erg,or
γ1?5×10?5 ?0.1 8/3,(10) consistent with the observed ratio Eγ/E rot=7×10?5for canonical values of M H=7M⊙and a rapidly spinning black hole(E rot=0.29M H).
In the suspended accretion state,the torus emits correlated energies in various channels, namely in gravitational radiation,torus winds and thermal and MeV-neutrino emissions. Their fractional energies,relative to the rotational energy of the black hole,satisfy(van Putten(2003),corrected and simpli?ed)
γ2=E gw
α(1+δ)+f2w～η,(11)
γ3=E w
α(1+δ)+f2w～η2,(12)
and,the fractional energy dissipated and converted mostly in MeV-neutrino emissions,
γ4=
E diss
that the strong viscosity limit satis?esη～1/(4α)in the limit asαbecomes large.These
results imply a torus temperature of about2MeV,whereby the dominant emission is in
MeV-neutrino emissions accompanied by subdominant thermal emissions.
The MeV nucleus is relativistically compact,whereby the dominant emission is in grav-
itational radiation,rather than electromagnetic radiation.Its compactness can be expressed
in terms of2π Egw0f gw dE,which expressed the amount of rotational energy relative to the linear size of the system,which is invariant under rescaling of the mass of the black hole
according the Kerr metric(Kerr1963).We have(van Putten(2001);van Putten&Levinson
(2002),updated with(13))
γ5=0.0035 η
7M⊙
3.65×1052erg 1/2
imparted by E r on the remnant envelope.That is,E SN?0.5βE w,whereby
E SN?2×1051erg β7M⊙ η
consistent with the partial explosion energy of about1050erg in ejecta with velocities in excess of0.5c,where c denotes the velocity of light(Li&Chevalier1999).Conversely,E k,iso can readily assume anomalously large values in excess of1052erg,depending on the degree of asphericity.
In our model,the explosion energies(17)represent normal SNe Ic values(H¨o?ich et al.1999).The term“hypernova”(Paczynski1998)applies only to the apparent energy E k,iso?2?3×1052erg in GRB980425(Iwamoto1998;Woosley et al.1999)upon assuming spherical geometry,not to the true kinetic energy E SN in the actual aspherical explosion.
As pointed out in§2,the GRB emissions are strongly anisotropic,produced by beamed baryon-poor jets along the rotational axis of the black hole.Based on consistency between the true GRB event rate,based on(Frail et al.2001;van Putten&Regimbau2003),and GRB980425,we further infer that these beamed emissions are accompanied by extremely weak gamma-ray emissions over wide angles or perhaps over all directions.The beaming factor of the baryon poor jet is about450(Frail et al.2001;van Putten&Regimbau2003). Evidently,the degree of anisotropy in the GRB emissions exceeds the axis ratio of2to3 in the associated supernova ejecta(H¨o?ich et al.1999)by about two orders of magnitude. While viewing the source on-axis gives rise to the brightest GRB and the largest E k,iso, we conclude that viewing the source o?-axis could give rise to an apparently dim GRB with nevertheless large E k,iso.This may explain the apparent discrepancy between the dim GRB980425in the presence of a large E k,iso,yet normal E SN(H¨o?ich et al.(1999),Eqn.17 above),in SN1998bw.
The remarkable similarity between the optical light-curve of SN2003dh associated with GRB030329(Stanek et al.2003)supports the notion that GRBs are driven by standard inner engines.GRB030329was a bright event in view of its proximity,though appeared with a slightly sub-energetic Eγ,iso.We attribute this to viewing strongly anisotropic GRB emissions slightly o?the rotational axis of the black hole.Based on spectral data,Kawabata et al.(2003)note that the energy E k,iso of SN2003dh is probably between that of SNe1997ef (e.g.Nomoto et al.(2001);Branch et al.(2001))and SN1998bw,although SN2003dh and SN1998bw feature similar initial expansion velocities.If SN2003dh allows a detailed aspher-ical model similar to that of SN1998bw,we predict that the true kinetic energy E SN will attain a normal value.
The observational constraint E SNR?2×1051erg on SN1998bw(H¨o?ich et al.1999) and consistency with the energy requirement in high-energy continuum emissions for the X-ray line-emissions in GRB011211,therefore,suggest an expectation value of f gw?500Hz according to(15)and(17).It would be of interest to re?ne this estimate by calorimetry on a sample of SNRs which are remnants of GRBs.Given the true GRB event of about
1per year within a distance of100Mpc,we anticipate about1GRB-SNR within10Mpc. These remnants will contain a black hole in a binary with an optical companion,possible representing a soft X-ray transient.
5.Line-broadening from Lense-Thirring precession
Quadrupole emissions in gravitational radiation emitted by the torus,possibly accom-panied by emissions from higher-order multipole mass-moments,represent a line,which changes on the secular timescale of the change in black hole-spin.This line will broaden, when the torus precesses.Lense-Thirring precession(Lense&Thirring1918;Wilkins1972) describes the e?ect of frame-dragging on a torus whose angular velocity vector is misaligned with the spin-axis of the black hole.Lense-Thirring precession is well-known in a di?er-ent context,as a possible mechanism for QPOs in X-ray binaries(Stella1999)as well as in black hole-neutron star binaries(Apostolatos et al.1994).A torus which is misaligned with the spin-axis of the black hole precesses with essentially the frame-dragging angular velocity described by the Kerr metric.This is accompanied by precession of the black hole, by conservation of angular momentum.Quite generally,the angular momentum of the torus is much less than that of the black hole,whereby the wobbling angle of the black hole is relatively small and can be neglected.
Precession of the orientation of the torus modulates its cosine with the line-of-sight. The observed strain-amplitudes are hereby phase-modulated.Phase-modulation of grav-itational radiation from an intrinsic quadrupole moment introduces line-broadening.For small phase-modulations,this is manifest in phase-coherent side-bands,which are separated from twice the orbital frequency by the frequency of Lense-Thirring precession.The ori-gin of a misaligned torus may result from misaligned spin-up of the progenitor star,prior to core-collapse,when the progenitor star is itself misaligned with the orbital plane of the binary.
In Boyer-Lindquist coordinates,we have to leading order the Lense-Thirring angular frequency?LT?2J H/R3for a black hole angular momentum J H=M2sinλin terms of the mass M and the speci?c angular momentum sinλ=a/M.Given the angular velocity ?H=tan(λ/2)/2M of the black hole and the angular velocity?T?M1/2R?3/2of the torus, we have
?LT
0.1 2sin2(λ/2)(18) in terms of the ratioη=?T/?H of the angular velocities of the torus to that of the black hole.We expect nominal valuesη～0.1(van Putten&Levinson2003),so that?LT is about
10%of?T,or,equivalently,about1%of?H.
An intrinsic mass-inhomogeneity m in a torus introduces a luminosity of gravitational radiation according to L gw=(32/5)(M/R)5(m/M)2,where M?M(m/M)3/5denotes the chirp mass.The gravitational radiation thus produced is anisotropic.For each of the two polarizations,we have
h+=4
2
cos(2?T t),h×=?
4
r
[cos(2?T+?LT)+cos(2?T??LT)](21)
and
h(1)×=
2sin(ι0)
1+6cos2ι0+cos4ι0 1/2sinι0,(23) where we used?LT<<2?T.Averaged over all anglesι0,we haveˉK?θ/2.Thus,a wobbling angle of about30o typically produces side-bands of relative strength20%(taking together h+and h×in each side-band).
The above shows that Lense-Thirring precession,if present,may introduce line-broadening by up to5%.
The same precession introduces time-harmonic modulation of the two principal projec-tions of the torus onto the celestial sphere,one at once and one at twice the precession
frequency.The strength of the two low-frequency lines de?nes the decay time of the mis-alignment of the torus.These lines are extremely small,in view of their low-frequencies, allowing Lense-Thirring to persist for timescales at least as long as the durations of long GRBs.
6.Stochastic background radiation from GRBs
We may calculate the contribution of GRBs from rotating black holes to the stochastic background in gravitational waves,for a distribution which is locked to the star-formation rate.Below is a semi-analytic summation of the sources,similar but not identical to the numerical summation in(Coward et al.2002),and includes a correction to the amplitudes reported therein.
The spectral energy density dE gw/d f of a single point source is a redshift-independent distribution,in view of Einstein’s adiabatic relationship E gw/f=const.The observed energy E gw(f,z)at an observed frequency f of a source at redshift z hereby satis?es E gw(f,z)= (1+z)?1E gw((1+z)f,0).Hence,we have E′gw(f,z)=E′gw((1+z)f,0)with′=d/d f.At redshift zero,gravitons emitted by a source at redshift z are distributed over a surface area 4πd2L(z),where d L(z)denotes the luminosity area.This gives rise to a spectral energy-density, or equivalently,a?ux per unit area at the observer,satisfying?F s(f,z)=E′gw/4πd2L(z). Given a star-formation rate R SF(z)as measured in the local rest frame per unit of comoving volume V at redshift z,the GRB event rate R as seen by the observer satis?es dR(z)/dz=˙n GRB(1+z)?1(R SF(z)/R SF(0))(dV/dz),where˙n GRB denotes the GRB rate-density at z=0. The result contributes to the spectral energy density,i.e.,?ux per unit area to the stochastic background in gravitons by
?F
(f)=˙n GRB z max0E′gwΣ(z)
B
(完整版)经济增加值eva计算方法
EVA计算方法 说明: 经济增加值(EVA)＝税后净营业利润（NOPAT）－资本成本（cost of capital） 资本成本＝资本×资本成本率 由上知，计算EVA可以分做四个大步骤：（1）税后净营业利润（NOPAT）的计算; (2)资本的计算；(3)资本成本率的计算；（4）EVA的计算。下面列出EVA的计算步骤，并以深万科（0002）为例说明EVA（2000年）的计算。 深万科（0002）简介： 公司名称：万科企业股份有限公司公司简称：深万科Ａ上市日期：1991-01-29 上市地点：上海证券交易所行业：房地产业股本结构：A 股398711877股，B股121755136 股，国有股、境内法人股共110504928股，股权合计数：630971941股。 一、税后净营业利润（NOPAT）的计算 1．以表格列出的计算步骤 下表中，最左边一列（以IS开头）代表损益表中的利润计算步骤，最右边一列（以NOPAT开头）代表计算EVA所用的税后净营业利润（NOPAT）的计算步骤。空格代表在计算相应指标（如NOPAT）的步骤中不包含该行所对应的项。
损益表中的利润计算步骤 税后净营业 利润 （NOPAT） 的计算步骤主营业务收入 - 销售折扣和折让- - 主营业务税金及附加- - 主营业务成本- 主营业务利润 + 其它业务利润+ 当年计提或冲销的坏帐准备+ - 当年计提的存货跌价准备 - 管理费用- - 销售费用- = 营业利润/调整后的营业利润 + 投资收益+
= 总利润/税前营业利润 - EVA税收调整* - = 净利润/税后净营业利润 2. 计算公式：(蓝色斜体代表有原始数据，紫色下划线代表此数据需由原始数据推算出) （1）税后净营业利润=主营业务利润＋其他业务利润＋当年计提或冲销的坏帐准备—管理费用—销售费用＋长期应付款，其他长期负债和住房公积金所隐含的利息＋投资收益—EVA税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润＝主营业务收入—销售折扣和折让—营业税金及附加—主营业务成本 注: 主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 (3)EVA税收调整=利润表上的所得税＋税率×（财务费用＋长期应
标准正态分布的密度函数样本
幻灯片1 正态分布 第二章 第七节 一、标准正态分布的密度函数 二、标准正态分布的概率计算 三、一般正态分布的密度函数 四、正态分布的概率计算幻灯片2 正态分布的重要性正态分布是概率论中最重要的分布, 这能够由 以下情形加以说明: ⑴ 正态分布是自然界及工程技术中最常见的分布之一, 大量的随机现象都是服从或近似服从正态分布的．能够证明, 如果一个随机指标受到诸多因素的影响, 但其中任何一个因素都不起决定性作用, 则该随机指标一定服从或近似服从正态分布． 这些性质是其它 ⑵ 正态分布有许多良好的性质, 许多分布所不具备的． ⑶ 正态分布能够作为许多分布的近似分布．幻灯片3 －标准正态分布下面我们介绍一种最重要的正态分布 一、标准正态分布的密度函数若连续型随机变量X 的密度函数为定义 则称X 服从标准正态分布,
记为标准正态分布是一种特别重要的它的密度函数经常被使用, 分布。 幻灯片4 密度函数的验证 则有 ( 2) 根据反常积分的运算有能够推出 幻灯片5 标准正态分布的密度函数的性质若随机变量 , X 的密度函数为 则密度函数的性质为: 的图像称为标准正态( 高斯) 曲线幻灯片6 随机变量 由于 由图像可知, 阴影面积为概率值。对同一长度的区间 , 若这区间越靠近 其对应的曲边梯形面积越大。标准正态分布的分布规律时”中间多, 两头少” . 幻灯片7 二、标准正态分布的概率计算 1、分布函数分布函数为幻灯片8 2、标准正态分布表书末附有标准正态分布函数数值表, 有了它, 能够解决标准正态分布的概率计算.表中给的是x > 0时，①(x)的值. 幻灯片9 如果由公式得令则幻灯片10
怎样理解分布函数
怎样理解分布函数 概率论中一个非常重要的函数就是分布函数,知道了随机变量的 分布函数,就知道了它的概率分布,也就可以计算概率了。 一、理解好分布函数的定义: F(x)=P(X≤x), 所以分布函数在任意一点x的值,表示随机变量落在x点左边(X≤x)的概率。它的定义域是(-∞,+∞),值域是[0,1]. 二、掌握好分布函数的性质: (1)0≤F(x)≤1; (2)F(+∞)=1,F(-∞)=0; 可以利用这条性质确定分布函数中的参数,例如: 设随机变量X的分布函数为:F(x)=A+Barctanx,求常数A与B. 就应利用本性质计算出A=1/2,B=1/π. (3)单调不减; (4)右连续性。 三、会利用分布函数求概率 在利用分布函数求概率时,以下公式经常利用。
(1)P(a 经济增加值(eva)计算方式 (四)（Economic value added (EVA) calculation (four)） Next, the calculation method of economic value added is introduced The calculation model of EVA is given below. Computational model of 1 and EVA Economic value added = net operating profit after tax - cost of capital = net operating profit after tax - total capital * weighted average cost of capital Among them: Net operating profit after tax net profit after tax interest expense + = + + minority income this year amortization of goodwill + deferred tax credit balances increase reserve balances increased + + other capitalized research and development costs, capitalized research and development costs in the years of amortization Total capital = common equity + minority interests + deferred tax credit (debit balance is negative) + + (cumulative amortization of goodwill reserve inventory impairment provision for bad debts, etc.) + + + capitalization amount of short-term loans for research and development costs of long term loan + short-term long-term loans due in part EVA 计算方法 说明: 经济增加值(EV A)＝税后净营业利润（NOPA T ）－资本成本（cost of capital ） 资本成本＝资本×资本成本率 由上知，计算EV A 可以分做四个大步骤： （1）税后净营业利润（NOPA T ）的计算; (2)资本的 计算； (3)资本成本率的计算； （4）EV A 的计算。下面列出EV A 的计算步骤，并以深万科（0002）为例说明EV A （2000年）的计算。 深万科（0002）简介： 公司名称：万科企业股份有限公司 公司简称：深万科Ａ 上市日期：1991-01-29 上市地点：上海证券交易所 行业：房地产业 股本结构：A 股398711877股，B 股121755136 股，国有股、境内法人股共110504928股，股权合计数：630971941股。 一、税后净营业利润（NOPA T ）的计算 1． 以表格列出的计算步骤 下表中，最左边一列（以IS 开头）代表损益表中的利润计算步骤，最右边一列（以NOPA T 开头）代表计算EV A 所用的税后净营业利润（NOPA T ）的计算步骤。空格代表在计算相应指标（如NOPA T ）的步骤中不包含该行所对应的项。 损益表中的利润计算步骤 税后净营业 利润 （NOPA T ）的计算步骤 主营业务收入 - 销售折扣和折让 - - 主营业务税金及附加 - - 主营业务成本 - 主营业务利润 - 管理费用 - - 销售费用 - = 营业利润/调整后的营业利润 + 投资收益 + = 总利润/税前营业利润 - EVA 税收调整* - = 净利润/税后净营业利润 2.计算公式：(蓝色斜体代表有原始数据，紫色下划线代表此数据需由原始数据推算出) （1）税后净营业利润=主营业务利润＋其他业务利润＋当年计提或冲销的坏帐准备—管理费用—销售费用＋长期应付款，其他长期负债和住房公积金所隐含的利息＋投资收益—EV A 税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润＝主营业务收入—销售折扣和折让—营业税金及附加—主营业务成本注: 主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 (3)EV A税收调整=利润表上的所得税＋税率×（财务费用＋长期应付款，其他长期负债 和住房公积金所隐含的利息＋营业外支出－营业外收入－补贴收入） (4)长期应付款，其他长期负债和住房公积金所隐含的利息＝长期应付款，其他长期负债 和住房公积金×3～5 年中长期银行贷款基准利率 长期应付款，其他长期负债和住房公积金=长期负债合计—长期借款—长期债券 税率=0.33（从1998年，1999年和2000年） 说明：上面计算公式所用数据大多直接可以在sternstewart公司所提供的原始财务数据中找到（主营业务利润已直接给出）。而长期应付款，其他长期负债和住房公积金所隐含的利息需由原始财务数据推算得出。 3. 计算深万科的税后净营业利润（NOPAT 2000年） 首先计算出需由其他原始财务数据推算的间接数据项－长期应付款，其他长期负债和住房公积金所隐含的利息和EV A税收调整，然后利用计算结果及其他数据计算出NOPA T. （1）长期应付款，其他长期负债和住房公积金所隐含的利息的计算； 单位：元 长期负债合计123895991.54 减：长期借款80000000.00 减：长期债券 ――――――――――――――――――――――――――――― 长期应付款，其他长期负债和住房公积金43895991.54 乘：3～5 年中长期银行贷款基准利率 6.03％ 长期应付款，其他长期负债2646928.29 和住房公积金所隐含的利息 （2）EV A税收调整的计算； 财务费用1403648.37 加：长期应付款，其他长期负债2646928.29 和住房公积金所隐含的利息 加：营业外支出6595016.31 减：营业外收入23850214.53 EV A计算方法 说明: 经济增加值(EV A)＝税后净营业利润（NOPAT）－资本成本（cost of capital） 资本成本＝资本×资本成本率 由上知，计算EV A可以分做四个大步骤： （1）税后净营业利润（NOPAT）的计算; （2）资本的计算； （3）资本成本率的计算； （4）EV A的计算。 下面列出EV A的计算步骤，并以深万科（0002）为例说明EV A（2000年）的计算。 深万科（0002）简介： 公司名称：万科企业股份有限公司公司简称：深万科Ａ上市日期：1991-01-29 上市地点：上海证券交易所行业：房地产业股本结构：A股398711877 股，B股121755136 股，国有股、境内法人股共110504928股，股权合计数：630971941股。一、税后净营业利润（NOPAT）的计算 1．以表格列出的计算步骤 下表中，最左边一列（以IS开头）代表损益表中的利润计算步骤，最右边一列（以NOPA T 开头）代表计算EV A所用的税后净营业利润（NOPA T）的计算步骤。空格代表在计算相 应指标（如NOPA T）的步骤中不包含该行所对应的项。 损益表中的利润计算步骤 税后净营业 利润 （NOPAT） 的计算步骤主营业务收入 - 销售折扣和折让- - 主营业务税金及附加- - 主营业务成本- 主营业务利润 - 管理费用- = 营业利润/调整后的营业利润 + 投资收益+ = 总利润/税前营业利润 = 净利润/税后净营业利润 2.计算公式：(蓝色斜体代表有原始数据，紫色下划线代表此数据需由原始数据推算出) （1）税后净营业利润=主营业务利润＋其他业务利润＋当年计提或冲销的坏帐准备—管理费用—销售费用＋长期应付款，其他长期负债和住房公积金所隐含的利息＋投资收益—EV A 税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润＝主营业务收入—销售折扣和折让—营业税金及附加—主营业务成本注: 主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 (3)EVA税收调整=利润表上的所得税＋税率×（财务费用＋长期应付款，其他长期负债 和住房公积金所隐含的利息＋营业外支出－营业外收入－补贴收入） (4)长期应付款，其他长期负债和住房公积金所隐含的利息＝长期应付款，其他长期负债 和住房公积金×3～5 年中长期银行贷款基准利率 长期应付款，其他长期负债和住房公积金=长期负债合计—长期借款—长期债券 税率=0.33（从1998年，1999年和2000年） 说明：上面计算公式所用数据大多直接可以在sternstewart公司所提供的原始财务数据中找到（主营业务利润已直接给出）。而长期应付款，其他长期负债和住房公积金所隐含的利息需由原始财务数据推算得出。 3. 计算深万科的税后净营业利润（NOPAT 2000年） 首先计算出需由其他原始财务数据推算的间接数据项－长期应付款，其他长期负债和住房公积金所隐含的利息和EV A税收调整，然后利用计算结果及其他数据计算出NOPA T. （1）长期应付款，其他长期负债和住房公积金所隐含的利息的计算； 单位：元 长期负债合计123895991.54 减：长期借款80000000.00 减：长期债券 ――――――――――――――――――――――――――――― 长期应付款，其他长期负债和住房公积金43895991.54 乘：3～5 年中长期银行贷款基准利率 6.03％ 长期应付款，其他长期负债2646928.29 和住房公积金所隐含的利息 （2）EV A税收调整的计算； 财务费用1403648.37 正态分布 第二章 第七节 一、标准正态分布的密度函数 二、标准正态分布的概率计算 三、一般正态分布的密度函数 四、正态分布的概率计算 幻灯片2 正态分布的重要性正态分布是概率论中最重要的分布， 这可以由 以下情形加以说明： ⑴正态分布是自然界及工程技术中最常见的分布 之一， 大量的随机现象都是服从或近似服从正态分布的． 可以证明， 如果一个随机指标受到诸多因素的影响, 但其中任何一个因素都不起决定性作用， 则该随机指标 一定服从或近似服从正态分布． 这些性质是其它 ⑵正态分布有许多良好的性质， 许多分布所不具备的． ⑶正态分布可以作为许多分布的近似分布． 幻灯片3 －标准正态分布 下面我们介绍一种最重要的正态分布 一、标准正态分布的密度函数 若连续型随机变量X的密度函数为 定义 则称X服从标准正态分布， 记为 标准正态分布是一种特别重要的 它的密度函数经常被使用， 分布。 幻灯片4 密度函数的验证 则有 （2）根据反常积分的运算有 可以推出 幻灯片5 标准正态分布的密度函数的性质 ，X的密度函数为 则密度函数的性质为： 的图像称为标准正态（高斯）曲线。 幻灯片6 随机变量 由于 由图像可知，阴影面积为概率值。 对同一长度的区间 ，若这区间越靠近 其对应的曲边梯形面积越大。 标准正态分布的分布规律时“中间多，两头少”. 幻灯片7 二、标准正态分布的概率计算 1、分布函数 分布函数为 幻灯片8 2、标准正态分布表 书末附有标准正态分布函数数值表，有了它，可以解决标准正态分布的概率计算. 表中给的是x > 0时, Φ(x)的值. 幻灯片9 如果 由公式得 令 则 幻灯片10 例1 解 幻灯片11 由标准正态分布的查表计算可以求得， 当X～N(0,1)时， 这说明，X 的取值几乎全部集中在[-3,3]区间内，超出这个范围的可能性仅占不到0.3%. 幻灯片12 三、一般正态分布的密度函数 如果连续型随机变量X的密度函数为 （其中 为参数） 的正态分布，记为 则随机变量X服从参数为 所确定的曲线叫 作正态(高斯）曲线. 幻灯片13 目录 1. 均匀分布 (1) 2. 正态分布（高斯分布） (2) 3. 指数分布 (2) 4. Beta分布（：分布） (2) 5. Gamm 分布 (3) 6. 倒Gamm分布 (4) 7. 威布尔分布（Weibull分布、韦伯分布、韦布尔分布） (5) 8. Pareto 分布 (6) 9. Cauchy分布（柯西分布、柯西-洛伦兹分布） (7) 2 10. 分布（卡方分布） (7) 8 11. t分布................................................ 9 12. F分布 ............................................... 10 13. 二项分布............................................ 10 14. 泊松分布（Poisson 分布）............................. 11 15. 对数正态分布........................................ 1. 均匀分布 均匀分布X ~U（a,b）是无信息的，可作为无信息变量的先验分布。 2. 正态分布（高斯分布） 当影响一个变量的因素众多，且影响微弱、都不占据主导地位时，这个变量 很可能服从正态分布，记作 X~N （」f 2）。正态分布为方差已知的正态分布 N （*2）的参数」的共轭先验分布。 1 空 f （x ）： —— e 2- J2 兀 o' E(X)， Var(X) _ c 2 3. 指数分布 指数分布X ~Exp （ ）是指要等到一个随机事件发生，需要经历多久时间。其 中，.0为尺度参数。指数分布的无记忆性： Plx s t|X = P{X t}。 f （X ）二 y o i E(X) 一 4. Beta 分布（一：分布） f （X ）二 E(X) Var(X)= (b-a)2 12 Var(X)二 1 ~2 EVA 计算方法 说明: 经济增加值（EVA ）=税后净营业利润（NOPAT ）—资本成 本（cost of capital ） 资本成本=资本x 资本成本率 由上知，计算EVA 可以分做四个大步骤： （1 ）税后净 营 业利润（NOPAT ）的计算;（2）资本的计算；（3）资本成本率的计算； （4） EVA 的计算。下面列出EVA 的计算步骤，并以深万科（0002 ） 为例说明EVA （2000年）的计算。 深万科（0002 ）简介： 公司名称：万科企业股份有限公司 A 上市日期：1991-01-29 股 398711877 股，B 股 121755136 股共110504928 股，股权合计数：630971941 股 一、税后净营业利润（NOPAT ）的计算 1 .以表格列出的计算步骤 下表中，最左边一列（以IS 开头）代表损益表中的利润计算步骤， 最右边一列（以NOPAT 开头）代表计算EVA 所用的税后净营业利 润（NOPAT ）的计算步骤。空格代表在计算相应指标（如NOPAT ） 的步骤中不包含该行所对应的 公司简称：深万科 上市地点：上海证券交 易所 行业：房地产业 股本结构：A 股，国有股、境内法人 项。 = 总利润/税前营业利润 -EVA税收调整* - 少数股东权益 = 净利润/税后净营业利润 2.计算公式：（蓝色斜体代表有原始数据，紫色下划线代表此数据需 由原始数据推算出） （1）税后净营业利润二主营业务利润+其他业务利润+当年计提或冲销的坏帐准备一管理费用一销售费用+长期应付款，其他长期负债和住房公积金所隐含的利息 +投资收益一EVA税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润=主营业务收入一销售折扣和折让一营业税金及附加一主营业务成本 注：主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 ⑶EVA税收调整二利润表上的所得税+税率x（财务费用+长期应 付款，其他长期负债和住房公积金所隐含的利息 +营业外支出- 营业外收 目录 1. 均匀分布 ...................................................................................................... 1 2. 正态分布（高斯分布） ........................................................................... 2 3. 指数分布 ...................................................................................................... 2 4. Beta 分布（β分布） .............................................................................. 2 5. Gamma 分布 .............................................................................................. 3 6. 倒Gamma 分布 ......................................................................................... 4 7. 威布尔分布(Weibull 分布、韦伯分布、韦布尔分布) ..................... 5 8. Pareto 分布 ................................................................................................. 6 9. Cauchy 分布（柯西分布、柯西-洛伦兹分布） (7) 10. 2χ分布（卡方分布） (7) 11. t 分布 ......................................................................................................... 8 12. F 分布 ........................................................................................................ 9 13. 二项分布 ................................................................................................ 10 14. 泊松分布（Poisson 分布） .............................................................. 10 15. 对数正态分布 ....................................................................................... 11 1. 均匀分布 均匀分布~(,)X U a b 是无信息的，可作为无信息变量的先验分布。 1 ()f x b a =- Generated by Foxit PDF Creator ? Foxit Software https://www.wendangku.net/doc/9510575110.html, For evaluation only. 经济增加值(EVA)计算方式(四) 八 下介绍经济增加值的计算方式 下面给出EVA的计算模式。 1、EVA的计算模型 经济附加值= 税后净营业利润—资本成本 = 税后净营业利润—资本总额* 加权平均资本成本 其中： 税后净营业利润= 税后净利润+ 利息费用+ 少数股东损益+ 本年商誉摊销+ 递延税项贷方余额的增加+ 其他准备金余额的增加+ 资本化研究发展费用—资本化研究发展费用在本年的摊销 资本总额= 普通股权益+ 少数股东权益+ 递延税项贷方余额（借方余额则为负值）+ 累计商誉摊销+ 各种准备金（坏帐准备、存货跌价准备等）+ 研究发展费用的资本化金额+ 短期借款+ 长期借款+ 长期借款中短期内到期的部分 加权平均资本成本= 单位股本资本成本+ 单位债务资本成本。 2、报表和账目的调整。 由于根据会计准则编制的财务报表对公司绩效的反映存在部分失真，在计算经济附加值时需要对一些会计报表科目的处理方法进行调整。 Stern Stewart财务顾问公司列出了160多项可能需要调整的会计项目，包括存货成本、重组费用、税收、营销费用、无形资产、货币贬值、坏帐准备金、重组费用以及商誉摊销等。但在考察具体企业时，一般一个企业同时涉及的调整科目不超过15项。但由于EVA是Stern Stewart财务顾问公司注册的商标，其具体的账目调整和运算目前尚没有对外公开。 （1）、单位债务资本成本 单位债务资本成本指的是税后成本，计算公式如下： 税后单位债务资本成本=税前单位债务资本成本*（企业所得税税率） 我国上市公司的负债主要是银行贷款，这与国外上市公司大量发行短期票据和长期债券的做法不同，因此可以以银行贷款利率作为单位债务资本成本。 根据有关研究，我国上市公司的短期债务占总债务的90%以上，由于我国的银行贷款利率尚未放开，不同公司贷款利率基本相同。因此，可用中国人民银行公布的一年期流动资金贷款利率作为税前单位债务资本成本，并根据央行每年调息情况加权平均。不同公司的贷款利率实际上存在一定差别，可根据自身情况进行调整。 （2）、单位股本资本成本 单位股本资本成本是普通股和少教股东权益的机会成本。通常根据资本资产价模型确定，计算公式如下： 普通股单位资本成本=无风险收益+β*市场组合的风险溢价 其中无风险利率可采用5年期银行存款的内部收益率。 国外一般以国债收益作为无风险收益表，我国的流通国债市场规模较小，居民的无风险投资以银行存款为主，因此以5年期银行存款的内部收益率代替。随着国债市场发展，将来也可以国债收益率为基准。 β 系数反映该公司股票相对于整个市场（一般用股票市场指数来代替）的系统风险，β系数越大，说明该公司股票相对于整个市而言风险越高，波动越大。 β值可通过公司股票收益率对同期股票市场指数（上证综指）的收益率回归计算得来。 市场组合的风险溢价反映整个证券市场相对于无风险收益率的溢价，目前有一些学者将我国的市场风险溢价定为4%。 （3）、研究发展费用和市场开拓费用 现行会计制度规定，公司必须在研究发展费用和市场开拓费用发生的当年将期间费用一次性予以核销。这种处理方法实际上否认了两种费用对企业未来成长所起的关键作用，而把它与一般的期间费用等同起来。 这种处理方法的一个重要缺点就是可能会诱使经营者减少对这两项费用的投入，这在效益不好的年份和管理人员即将退休的前几年尤为明显。美国的有关研究表明，当管理人员临近退休之际，研究发展费用的增长幅度确实有所降低。 图 6-2 正态分布概率密度函数的曲线 正态曲线可用方程式表示。当n→∞时，可由二项分布概率函数方程推导出正态分布曲线的方程： f(x)= （6.16 ） 式中： x —所研究的变数； f(x) —某一定值 x 出现的函数值，一般称为概率密度函数（由于间断性分布已转变成连续性分布，因而我们只能计算变量落在某一区间的概率，不能计算变量取某一值，即某一点时的概率，所以用“概率密度”一词以与概率相区分），相当于曲线 x 值的纵轴高度； p —常数，等于 3.14 159 ……； e —常数，等于 2.71828 ……；μ为总体参数，是所研究总体的平均数，不同的正态总体具有不同的μ，但对某一定总体的μ是一个常数；δ也为总体参数，表示所研究总体的标准差，不同的正态总体具有不同的δ，但对某一定总体的δ是一个常数。 上述公式表示随机变数 x 的分布叫作正态分布，记作 N( μ , δ2 ) ，读作“具平均数为μ，方差为δ 2 的正态分布”。正态分布概率密度函数的曲线叫正态曲线，形状见图 6-2 。 （二）正态分布的特性 1 、正态分布曲线是以 x= μ为对称轴，向左右两侧作对称分布。因的数值无论正负，只要其绝对值相等，代入公式（ 6.16 ）所得的 f(x) 是相等的，即在平均数μ的左方或右方，只要距离相等，其 f(x) 就相等，因此其分布是对称的。在正态分布下，算术平均数、中位数、众数三者合一位于μ点上。 2 、正态分布曲线有一个高峰。随机变数 x 的取值范围为（ - ∞，+ ∞ ），在（ - ∞ ，μ）正态曲线随 x 的增大而上升，；当 x= μ时， f(x) 最大；在（μ，+ ∞ ）曲线随 x 的增大而下降。 3 、正态曲线在︱x-μ︱=1 δ处有拐点。曲线向左右两侧伸展，当x →± ∞ 时，f(x) →0 ，但 f(x) 值恒不等于零，曲线是以 x 轴为渐进线，所以曲线全距从 -∞到+ ∞。 4 、正态曲线是由μ和δ两个参数来确定的，其中μ确定曲线在 x 轴上的位置 [ 图 6-3] ，δ确定它的变异程度 [ 图 6-4] 。μ和δ不同时，就会有不同的曲线位置和变异程度。所以，正态分布曲线不只是一条曲线，而是一系列曲线。任何一条特定的正态曲线只有在其μ和δ确定以后才能确定。 5 、正态分布曲线是二项分布的极限曲线，二项分布的总概率等于 1 ，正态分布与 x 轴之间的总概率（所研究总体的全部变量出现的概率总和）或总面积也应该是等于 1 。而变量 x 出现在任两个定值 x1到x2(x1≠x2)之间的概率，等于这两个定值之间的面积占总面积的成数或百分比。正态曲线的任何两个定值间的概率或面积，完全由曲线的μ和δ确定。常用的理论面积或概率如下： 区间μ ± 1 δ面积或概率 =0.6826 μ ± 2 δ =0.9545 μ ± 3 δ=0.9973 μ± 1.960δ=0.9500 μ ±2.576 δ =0.9900 国资委经济增加值考核细则 （详细资料 https://www.wendangku.net/doc/9510575110.html,/flfg/2010-01/22/content_1517096.htm） 一、经济增加值的定义及计算公式 经济增加值是指企业税后净营业利润减去资本成本后的余额。 计算公式： 经济增加值=税后净营业利润-资本成本=税后净营业利润-调整后资本×平均资本成本率 税后净营业利润=净利润+（利息支出+研究开发费用调整项-非经常性收益调整项×50%）×（1-25%） 调整后资本=平均所有者权益+平均负债合计-平均无息流动负债-平均在建工程 二、会计调整项目说明 （一）利息支出是指企业财务报表中“财务费用”项下的“利息支出”。 （二）研究开发费用调整项是指企业财务报表中“管理费用”项下的“研究与开发费”和当期确认为无形资产的研究开发支出。对于为获取国家战略资源，勘探投入费用较大的企业，经国资委认定后，将其成本费用情况表中的“勘探费用”视同研究开发费用调整项按照一定比例（原则上不超过50%）予以加回。 （三）非经常性收益调整项包括 1.变卖主业优质资产收益：减持具有实质控制权的所属上市公司股权取得的收益（不包括在二级市场增持后又减持取得的收益）；企业集团（不含投资类企业集团）转让所属主业范围内且资产、收入或者利润占集团总体10%以上的非上市公司资产取得的收益。 2.主业优质资产以外的非流动资产转让收益：企业集团（不含投资类企业集团）转让股权（产权）收益，资产（含土地）转让收益。 3.其他非经常性收益：与主业发展无关的资产置换收益、与经常活动无关的补贴收入等。 （四）无息流动负债是指企业财务报表中“应付票据”、“应付账款”、“预收款项”、“应交税费”、“应付利息”、“其他应付款”和“其他流动负债”；对于因承担国家任务等原因造成“专项应付款”、“特种储备基金”余额较大的，可视同无息流动负债扣除。 （五）在建工程是指企业财务报表中的符合主业规定的“在建工程”。 三、资本成本率的确定 （一）中央企业资本成本率原则上定为5.5%。 （二）承担国家政策性任务较重且资产通用性较差的企业，资本成本率定为4.1%。 （三）资产负债率在75%以上的工业企业和80%以上的非工业企业，资本成本率上浮0.5个百分点。 （四）资本成本率确定后，三年保持不变。 四、其他重大调整事项 发生下列情形之一，对企业经济增加值考核产生重大影响的，国资委酌情予以调整。 （一）重大政策变化； （二）严重自然灾害等不可抗力因素； （三）企业重组、上市及会计准则调整等不可比因素 目录 1.均匀分布 (1) 2.正态分布（高斯分布） (2) 3.指数分布 (2) 4.Beta分布（β分布） (2) 5.Gamma分布 (3) 6.倒Gamma分布 (4) 7.威布尔分布(Weibull分布、韦伯分布、韦布尔分布) (5) 8.Pareto分布 (6) 9.Cauchy分布（柯西分布、柯西-洛伦兹分布） (7) χ分布（卡方分布） (7) 10.2 11.t分布 (8) 12.F分布 (9) 13.二项分布 (10) 14.泊松分布（Poisson分布） (10) 15.对数正态分布 (11) 1.均匀分布 均匀分布~(,) X U a b是无信息的，可作为无信息变量的先验分布。 1()f x b a = - ()2 a b E X += 2 ()()12 b a Var X -= 2. 正态分布（高斯分布） 当影响一个变量的因素众多，且影响微弱、都不占据主导地位时，这个变量很可能服从正态分布，记作2~(,)X N μσ。正态分布为方差已知的正态分布 2(,)N μσ的参数μ的共轭先验分布。 22 ()2()x f x μσ-- = ()E X μ= 2()Var X σ= 3. 指数分布 指数分布~()X Exp λ是指要等到一个随机事件发生，需要经历多久时间。其中0λ>为尺度参数。指数分布的无记忆性：{}|{}P X s t X s P X t >+>=>。 (),0 x f x e x λλ-=> 1 ()E X λ = 2 1 ()Var X λ = 4. Beta 分布（β分布） Beta 分布记为~(,)X Be a b ，其中Beta(1,1)等于均匀分布，其概率密度函数可凸也可凹。如果二项分布(,)B n p 中的参数p 的先验分布取(,)Beta a b ，实验数据（事件A 发生y 次，非事件A 发生n-y 次），则p 的后验分布(,)Beta a y b n y ++-，即Beta 分布为二项分布(,)B n p 的参数p 的共轭先验分布。 10 ()x t x t e dt ∞--Γ=? 1 1()()(1)()() a b a b f x x x a b --Γ+= -ΓΓ ()a E X a b = + 2 ()()(1) ab Var X a b a b = +++ 5. Gamma 分布 Gamma 分布即为多个独立且相同分布的指数分布变量的和的分布，解决的 正态分布的数学期望与方差 正态分布： 密度函数为：分布函数为 的分布称为正态分布，记为N(a, σ2). 密度函数为： 或者 称为n元正态分布。其中B是n阶正定对称矩阵，a是任意实值行向量。 称N(0,1)的正态分布为标准正态分布。 （1）验证是概率函数（正值且积分为1） （2）基本性质： （3）二元正态分布： 其中， 二元正态分布的边际分布仍是正态分布： 二元正态分布的条件分布仍是正态分布： 即（其均值是x的线性函数） 其中r可证明是二元正态分布的相关系数。 （4）矩，对标准正态随机变量，有 （5）正态分布的特征函数 多元正态分布 （1）验证其符合概率函数要求（应用B为正定矩阵，L为非奇异阵，然后进行向量线性变换） （2）n元正态分布结论 a) 其特征函数为： b) 的任一子向量,m≤n 也服从正态分布，分布为其中，为保留B 的第，…行及列所得的m阶矩阵。 表明：多元正态分布的边际分布还是正态分布 c) a,B分别是随机向量的数学期望及协方差矩阵，即 表明：n元正态分布由它的前面二阶矩完全确定 d) 相互独立的充要条件是它们两两不相关 e) 若，为的子向量，其中是，的协方差矩阵，则是，相应分量的协方差构成的相互协方差矩阵。则相互独立的充要条件为＝0 f) 服从n元正态分布N(a,b)的充要条件是它的任何一个线性组合服 从一元正态分布 表明：可以通过一元分布来研究多元正态分布 g) 服从n元正态分布N(a,b),C为任意的m×n阶矩阵，则服从m元正态分布 表明：正态变量在线性变换下还是正态变量，这个性质简称正态变量的线性变换不变性 推论：服从n元正态分布N(a,b)，则存在一个正交变化U，使得是一个具有独立正态分布分量的随机向量，他的数学期望为Ua，而他的方差分量是B的特征值。 条件分布 若服从n元正态分布N(a,b)，，则在给定下，的分布还是正态分布，其条件数学期望： (称为关于的回归) 其条件方差为： （与无关） Excel中的正态分布的密度函数 关于在Excel中的正态分布的密度函数NORMDIST（x，μ，σ，逻辑值）中积累逻辑值取“FALSE”时的图形，在《Excel：正态分布的面积图（积累逻辑值为FALSE）》（地址见【附录】）中简单作了尝试。现为了绘制正态累计分布逻辑值要取“TRUE”。 在Excel中的正态分布的密度函数NORMDIST的语法表达式是： NORMDIST(值，平均数，标准差，积累与否)，其中： x ——“值”，是要求分布的随机变量数值； μ——“平均数”，是分布的算数平均数； σ——“标准差”，是分布的标准差； 逻辑值——“积累与否”，是决定函数的逻辑值，其中： ●如果“积累与否”的逻辑值取“TRUE”（真），则NORMDIST 会返回累计分布函数。如果为了绘制正态累计分布，逻辑值就要取 “TRUE”。 ●如果“积累与否”的逻辑值取“FALSE”（伪），则NORMDIST 会返回正态分布函数的高度。 为了制作正态累计分布面积图，先准备下列数据表格（实际使用的表格中，单元格中都是数据，以下为了说明具体公式，在“工具”-“选项”-“视图”中勾选了“公式”，以便各单元格的具体参数都显示出来，以供参考。实际使用时还应该将这个勾选取消）。下列表格中各列NORMDIST函数中的逻辑值都取“TRUE”： 表1 在A列，准备按自己需要设置自变量数据x，本例从0——100，（A2——A102）。 在F列：B2=NORMDIST(A2,50,5, TRUE)，μ=50，σ=5，一直拖到F102。 在G列：G2=NORMDIST(A2,50,10, TRUE)，μ=50，σ=10，一直拖到G102。 在H列：H2=NORMDIST(A2,50,15, TRUE)，μ=50，σ=15，一直拖到H102。 在I列：I2=NORMDIST(A2,70,8, TRUE)，μ=70，σ=8，一直拖到I102。 先选取I列，选取I2：I102，作二维面积图，如图1所示： 图1 再选取H列，选取H2：H102，作二维面积图，如图2所示：经济增加值(eva)计算方式 (四)(Economic value added (EVA) calculation (four))
经济增加值EVA计算方法
经济增加值EVA的计算方法
标准正态分布的密度函数
16种常见概率分布概率密度函数、意义及其应用
经济增加值eva计算方法
16种常见概率分布概率密度函数、意义及其应用
正态分布概率公式(部分)
图 62正态分布概率密度函数的曲线 正态曲线可用方程式表示。 n 当 →∞时，可由二项分布概率函数方程推导出正态 分布曲线的方程：
fx= （61 ） () .6
式中： x—所研究的变数； fx —某一定值 x出现的函数值，一般称为概率 () 密度函数 （由于间断性分布已转变成连续性分布，因而我们只能计算变量落在某 一区间的概率， 不能计算变量取某一值， 即某一点时的概率， 所以用 “概率密度” 一词以与概率相区分），相当于曲线 x值的纵轴高度； p—常数，等于 31 .4 19……； e— 常数，等于 2788……； μ 为总体参数，是所研究总体 5 .12 的平均数， 不同的正态总体具有不同的 μ ， 但对某一定总体的 μ 是一个常数； δ 也为总体参数， 表示所研究总体的标准差， 不同的正态总体具有不同的 δ ， 但对某一定总体的 δ 是一个常数。 上述公式表示随机变数 x的分布叫作正态分布， 记作 N μ ,δ2 )， “具 ( 读作 2 平均数为 μ，方差为 δ 的正态分布”。正态分布概率密度函数的曲线叫正态 曲线，形状见图 62。 （二）正态分布的特性
1、正态分布曲线是以 x μ 为对称轴，向左右两侧作对称分布。因 =
的
数值无论正负， 只要其绝对值相等， 代入公式 61 ） （ .6 所得的 fx 是相等的， () 即在平均数 μ 的左方或右方，只要距离相等，其 fx 就相等，因此其分布是 () 对称的。在正态分布下，算术平均数、中位数、众数三者合一位于 μ 点上。经济增加值(EVA)计算方式 (四) 八
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