每周一题

《每周一题》 题目与答案

[第1周] 在一次乒乓球比赛中设立奖金1000元.比赛规定谁先胜了三盘,谁获得全部奖金.设甲,乙二人的球技相等,现已打了3盘,甲两胜一负,由于某种特殊的原因必需中止比赛.问这1000元应如何分配才算公平?

[第2周] 已知,4/1)()()(===C P B P A P ,0)(=AB P ,6/1)()(==BC P AC P 则事件C B A ,,全不发生的概率为 . 通过做此题你能发现什么问题？（此题为1992年工科生考研题）

[第3周] 17世纪来,法国的Chevalies De Mere 注意到在赌博中一对骰子抛25次,把赌注押到”至少出现一次双六”比把赌注押到”完全不出现双六”有利.但他本人找不出原因.后来请当时著名的法国数学家Pascal 才解决了这一问题.这问题是如何解决的呢?

[第4周] 某市进行艺术体操赛, 需设立两个裁判组, 甲组3名,乙组1名. 但组委会只召集 到3名裁判, 由于临近比赛, 便决定调一名不懂行的行政干部参加甲组工作, 其中两裁判独立地以概率 p 作出正确裁定,而行政干部以掷硬币决定, 最后根据多数人的意见决定.乙组由 1 个人组成, 他以概率 p 做出正确裁定. 问哪一组做出正确裁定的概率大 ?

[第5周] 自动生产线调整以后出现废品的概率为 p , 当生产过程中出现废品时立即重新进行调整, 求在两次调整之间的合格产品数的分布.

[第6周] 在高为 h 的三角形ABC 中任取一点M , 点 M 到 AB 的距离为随机变量X , 求其密度函数 f (x ).

[第7周] 某中外合资公司准备通过考试招工200名，其中180名正式工，20名临时工. 报考人数为1684名，考试满分为300分. 阅卷后人事部门公布如下信息：平均成绩是178分， 270以上的高分有32名. 考生小王成绩是233分，他能否被录取？如被录取能否是正式工？

[第8周] 设随机变量 服从参数为1的指数分布, 引入随机变量：

.

求二维随机变量 的联合分布律和联合分布函数 .

[第9周] 设随机变量 与 相互独立，其中 .

求随机变量 的概率密度函数 .

[第10周] 某民营企业生产的某产品每周的需求量 X (单位: 箱) 取[1 , 5]上的每个整数值是等可能的. 生产每箱产品的成本是300元,出厂价每箱900元.若售不出, 则每箱以100元的保管费借冷库保存. 问该企业每周生产多少产品能使获利的期望值最大？(精确到一位小数)

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(~y f Y Z X Y ,1110???>≤=Z Z X ???>≤=212

Z Z Y ),(Y X )

,(y x F )(z g Y X Z -=3

[第11周] 电视台需作节目A 收视率的调查. 每天在播电视的同时, 随机地向当地居民打电话询问是否在看电视. 若在看电视, 再问是否在看节目A. 设回答看电视的居民户数为n. 若要保证以95%的概率使调查误差在10%之内, n 应取多大？若使调查误差在1%之内, n 又取多大？

每晚节目A 播出一小时, 调查需同时进行, 设每小时每人能调查20户, 每户居民每晚看电视的概率为70%, 电视台需安排多少人作调查.

[第12周] 一本书有1 000 000 个印刷符号, 排版时每个符号被排错的概率为千分之一. 校对时, 每个排版错误被改正的概率为0.99. 求在校对后错误不多于15个的概率.

[第13周] 如何估计湖中黑、白鱼的比例

某水产养殖场两年前在人工湖中混养了黑、白两种鱼. 现在需要对黑白鱼数目的比例进行估计.

提示：分别用矩法与极大似然估计法解决此问题.

[第14周] 母亲嗜酒是否影响下一代的健康

美国的Jones医生于1974年观察了母亲在妊娠时曾患慢性酒精中毒的6名七岁儿童（称为甲组）.以母亲的年龄，文化程度及婚姻状况与前6名儿童的母亲相同或相近，但不饮酒的46名七岁儿童为对照租(称为乙组). 测定两组儿童的智商，结果如下：

智商人数智商平均数样本标准差

组别n x s

甲组 6 78 19

乙组46 99 16

由此结果推断母亲嗜酒是否影响下一代的智力？若有影响，推断其影响程度有多大?

提示：前一问题属假设检验问题，后一问题属区间估计问题.

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