【统计学】第五章

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思考题

1.频率与概率有什么关系？

事件A的概率是描述事件A在试验中出现的可能性的大小的一种度量，记事件A出现的可能性大小的数值为P(A),P(A)称为事件A的概率。
频率更接近实验值，概率更接近理论值。

2.独立性与互斥性有什么关系？

独立性：两个事件不论哪一个事件发生并不影响另一个事件发生的概率，则称这两个事件相互独立。
互斥性：互斥事件一定是相互依赖(不独立)的，但是相互依赖的事件不一定是互斥的。
不互斥事件可能是独立的，也可能是不独立的，然而独立事件不可能是互斥的

3.根据自己的经验体会举几个服从泊松分布的随机变量的实例

1.在某企业中每月发生的事故的次数
2.单位时间内到达某一服务柜台(服务站、诊所、超市的收银台、电话总机等)需要服务的顾客人数
3.人寿保险公司每天收到的死亡声明的个数
4.某种仪器每月出现故障的次数

4.根据自己的经验体会举几个服从正态分布的随机变量的实例

1.某地区同年龄组儿童的发育特征，如身高、体重、肺活量
2.某公司年销售量
3.在同一条件下产品的质量以平均质量为中心上下摆动

练习题

写出下列随机试验的样本空间

1.记录某班一次统计学测验的平均分数
某班人数
2.某人在公路上骑自行车，观察该骑车人在遇到第一个红灯停下来以前遇到的绿灯次数
所有灯
3.生产产品，直到有10件正品为止，记录生产产品的总件数
产品总数

某市有50%的住户订日报，有65%的住户订晚报，有85%的住户至少订两种报纸中的一种，求同时订这两种报纸的住户的百分比。

0.5+0.65−0.85=0.3

设A与B是两个随机事件，已知A与B至少有一个发生的概率是1/3，A发生且B不发生的概率是1/9，求B发生的概率

P(B)=31​−91​=92​

有甲、乙两批种子，发芽率分别是0.8和0.7。在两批种子中随机抽取一粒，求1.两粒都发芽的概率；2.至少有一粒发芽的概率；3.恰有一粒发芽的概率

1.0.56
2.0.94
3.0.38

某厂产品的合格率为96%，合格品中一级品率为75%，从产品中任取一件为一级品的概率为？

P(AB)=P(A)⋅P(A∣B)=0.96∗0.75=0.72

某种品牌的电视机用到5000小时未坏的概率为3/4，用到10000小时未坏的概率为1/2.现在有一台这种品牌的电视已经用了5000小时未坏，它能用到10000小时的概率？

P(A)=21​P(AB)=43​P(B)=32​

考虑掷两枚硬币的试验。令X表示观察到正面的个数，试求X的概率分布。

X	结果	概率
2	正正	0.25
1	正反	0.25
1	反正	0.25
0	反反	0.25

某人花2元钱买彩票，他抽中100元奖的概率是0.1%，抽中10元奖的概率是1%，抽中1元奖的概率是20%，假设各种奖不能同时抽中，试求

1.此人收益的概率分布

2.此人收益的期望值

结果	概率
抽中100元	0.1%
抽中10元	1%
抽中1元	20%

期望值：0.4

设随机变量X的概率密度为：

f(x)=θ33x2​,0<x<θ

(1)已知$P(X>1)=\frac{7}{8}%,求 θ的值
(2) 求X的期望值和方差

87​=∫1θ​θ33x2​dx

θ=2

E(x)=∫02​x83x2​dx=23​

E(x)=∫02​x283x2​dx=512​

D(x)=Ex2−(Ex)2=203​

一张考卷上有5道题目，同时每道题列出4个备选答案，其中有一个答案是正确的。某学生凭猜测能答对至少4道题的概率是多少？

41​5+5∗43​∗41​4=641​

设随机变量X服从参数为λ的泊松分布，且已知P{X=1}=P{X=2},求P{X=4}

泊松分布公式

P(X=k)=k!λk​e−λ

λ=2带入公式

PX=4=32​e−2

一工厂生产的电子管寿命X(以小时计算)服从期望值μ=160的正态分布，若要求P{120<X<200}>=0.08,允许标准差最大为多少？

工厂生产的电子管寿命在120＜X＜200之间时的概率为 P{120＜X＜200} ＝ P{(120－160)／δ＜ (X－120)／δ＜ (200－160)／δ} ＝ P{-40／δ＜ Z ＜40／δ} ＝φ(40／δ)－φ(-40／δ) ＝ 2φ(40／δ)－1 ≥ 0.08 →φ(40／δ) ≥0.54 通过查阅标准正态分布表得 40／δ≥0.11 δ≤363.64

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