2025 春季学期 概率论与数理统计 期中考试试卷

SOUTHERN UNIVERSITY OF SCIENCE AND TECHNOLOGY

考试科目: 概率论与数理统计

考试时间: 2025/4/19 19: 00-21: 00

开课单位: 数学系

命题教师: 概率统计教学组

本试卷共三大部分, 满分(100) 分 (考试结束后请将试卷, 答题卡一起交给监考老师)

题号	I	II	III-1	III-2	III- 3	III-4	III-5	III-6
分值	20 分	20 分	10 分	10 分	10 分	10 分	10 分	10 分

1 选择题(每题4分, 总共20分)

Part I-Single Choice(4 marks each question, 20 marks in total)

1-1

将 7 个人分成 3 组来执行相同的任务, 其中一组有 3 个人, 另外两组都有 2 个人.有多少种分组方式?

Divide 7 people into 3 groups to perform the same task, with one group of 3 and the other two groups of 2 each.How many ways are there to form the groups?

A. 35

B. 210

C. 105

D. 70

1-2

随机事件 $A$ 和 $B$ 满足 $P(A)=P(B)=\frac{1}{2}$ 且 $P(A\cup B)=1$ , 那么以下哪一项是正确的? Suppose random events $A$ and $B$ satisfy $P(A)=P(B)=\frac{1}{2}$ and $P(A\cup B)=1$ , then which of the following is true?

A.$A\cup B=\mathrm{\Omega }$

B.$P(A-B)=0$

C.$P(A\cup \overline{B})=1$

D.$A\cap B=\mathrm{\varnothing }$

1-3

某种医学检测会给出阳性或阴性结果. 阳性结果旨在表明一个人患有某种(罕见) 疾病, 而阴性结果旨在表明他们没有该疾病. 然而, 假设该检测有时会给出错误的结果: 在没有该疾病的 4 中, 有 $1/100$ 的人会得到阳性检测结果；而在患有该疾病的人中, 有 $2/100$ 的人会得到阴性检测结果.如果每 1000 人中有 1 人患有该疾病, 那么一个检测结果为阳性的人实际上患有该疾病的概率是?

A certain medical test either gives a positive or negative result. The positive test result is intended to indicate that a person has a particular (rare) disease, while a negative test result is intended to indicate that they do not have the disease. Suppose, however, that the test sometimes gives an incorrect result: 1 in 100 of those who do not have the disease have positive test results, and 2 in 100 of those having the disease have negative test results. If 1 person in 1000 has the disease, find the probability that a person with a positive test result actually has the disease.

A.$\frac{1097}{10000}$

B.$\frac{98}{1097}$

C.$\frac{1001}{10000}$

D.$\frac{98}{1007}$

1-4

随机变量 $X$ 的密度函数为 $f_X(x)=\{\begin{array}{ll}1-|x|,& -1\le x\le 1\\ 0,& \text{ 其他 }\end{array}$ . 求其分布函数在 $x=\frac{1}{2}$处的值.

The probability density function of the random variable $X$ is $f_X(x)=1-|x|$ for $-1\le x\le 1$. Find the value of its distribution function at $x=\frac{1}{2}$.

A.$\frac{1}{8}$

B.$\frac{3}{8}$

C.$\frac{5}{8}$

e. $\frac{7}{8}$

D. $\frac{7}{8}$

1-5

给定联合密度函数 $f(x,y)=\{\begin{array}{ll}{e}^{-y},& 0<x<y<\mathrm{\infty }\\ 0,& \text{ 其他 }\end{array}$ . 以下哪一项是正确的?

Given the joint density function $f(x,y)=\{\begin{array}{ll}{e}^{-y},& 0<x<y<\mathrm{\infty }\\ 0,& \text{ otherwise }\end{array}$. Which one of the following is true?

A. $f(3,2)=e^{-2}$

B. $f_X(x)f_Y(y)=f(x,y)$

C. $f_{Y\mid X}(y\mid x)=\{\begin{array}{ll}{e}^{-(y-x)},& y>x>0,\\ 0,& \text{ otherwise }.\end{array}$

D. $f_X(x)=\{\begin{array}{ll}x{e}^{-x},& x>0,\\ 0,& \text{ otherwise.}\end{array}$

2 填空题(每空2分, 总共20分)

Part II-Blank Filling(2 marks each blank, 20 marks in total)

2-1

设 $A_1,A_2,A_3$ 是样本空间的一个划分且 $P\left(A_1\right)=0.2,P\left(A_2\right)=0.5,P(B\mid A_1)=0.1$ , $P(B\mid A_2)=0.4,P(B\mid A_3)=0.7$ .那么 $P(A_3\mid B)=$ $\underset{―}{}$

Let $A_1,A_2,A_3$ be a partition of the sample space, with: $P\left(A_1\right)=0.2,P\left(A_2\right)=0.5$ , $P(B\mid A_1)=0.1,P(B\mid A_2)=0.4$ , and $P(B\mid A_3)=0.7$ . Then $P(A_3\mid B)=$ $\underset{―}{}$

2-2

一个不透明的袋子中装有5红球和3白球, 每次取1球不放回, 直到取出所有白球为止.设停止时总取球次数为 $X$ , 则 $P(X=6)=$ $\underset{―}{}$ . An opaque bag contains 5 red balls and 3 white balls. Balls are drawn one by one without replacement until all white balls are taken out. Let $X$ be the total number of draws when the process stops. Then, the probability $P(X=6)=$ $\underset{―}{}$

2-3

在长度为 $a$ 的线段上, 于中点两侧各随机取一点, 将线段分成三段.那么这三段长度不能构成三角形的概率是 $\underset{―}{}$.

On a line segment of length $a$ , two points are selected independently and uniformly at random, one on each side of the midpoint.This divides the segment into three parts. Then the probability that these three lengths cannot form a triangle is $\underset{―}{}$

2-4

设随机变量 $Y$ 服从参数为 1 的指数分布, $a$ 为常数且大于 0 , 则 $P\{Y\ge a+1\mid Y>a\}=$ $\underset{―}{}$ . Assume the random variable $Y$ follows exponential distribution with the parameter 1. If $a$ is a constant being greater than 0 , then $P\{Y\ge a+1\mid Y>a\}=$ $\underset{―}{}$

2-5

假设离散型随机变量 $X$ 的频率函数为 $P(X=1)=0.6,P(X=0)=0.4$ 且当 $X=1$ 时, $Y\sim U(0,2)$ ；当 $X=0$ 时, $Y\sim \mathrm{Exp}(1)$ .那么 $P(Y\le 1)=$ $\underset{―}{}$ . Assume that the frequency function of discrete random variable $X$ is $P(X=1)=0.6$ , $P(X=0)=0.4$ . Additionally, we have that if $X=1$ then $Y\sim U(0,2)$ ；and if $X=0$ then $Y\sim \mathrm{Exp}(1)$ . Then $P(Y\le 1)=$ $\underset{―}{}$

2-6

设 $(X,Y)$ 的联合密度为 $f_{X,Y}(x,y)=6e^{-x}{e}^{-2y}(0<x<y<+\mathrm{\infty })$ .则给定 $X=x$ 时, $Y$ 的条件密度为 $f_{Y\mid X}(y\mid x)=$ $\underset{―}{}$ $(y>x)$ . Let $(X,Y)$ have the joint probability density function: $f_{X,Y}(x,y)=6e^{-x}{e}^{-2y}$ for $0<x<y<+\mathrm{\infty }$ . Then the conditional density is $f_{Y\mid X}(y\mid x)=$ $\underset{―}{}$ $(y>x)$

2-7

假设 $X\sim N(\mu ,4)$ 为一个正态分布随机变量.已知 $P(X\le 1)=0.1587$ , 则 $\mu =$ $\underset{―}{}$ (提示: $\mathrm{\Phi }(1)=0.8413$ , 其中 $\mathrm{\Phi }$ 为标准正态分布函数)

2-8

设 $(X,Y)$ 的联 合 密 度 函 数 为 $f(x,y)=\{\begin{array}{ll}k,& 0<x^2<y<x<1\\ 0,& \text{ 其他 }\end{array}$ .则 常 数 $k=$ $\underset{―}{}$

2-9

设随机变量 $X$ 与 $Y$ 相互独立, $X$ 取值为 1 和 2 的概率均为 $0.5,Y$ 取值为 0 和 1 的概率均为0.5.令 $Z=X+Y$ , 则 $P(Z=2)=$ $\underset{―}{}$ .

2-10

设 $X$ 与 $Y$ 是两个随机变量, 且 $P\{X\ge 0,Y\ge 0\}=\frac{3}{7},P\{X\ge 0\}=P\{Y\ge 0\}=\frac{4}{7}$ .则 $P\{max\{X,Y\}\ge 0\}=$ $\underset{―}{}$ . Assume that $X$ and $Y$ are two random variables, and $P\{X\ge 0,Y\ge 0\}=\frac{3}{7};P\{X\ge 0\}=P\{Y\ge 0\}=\frac{4}{7}$ . Then $P\{max\{X,Y\}\ge 0\}=$ $\underset{―}{}$

3 解答题(每题10分, 共60分)

Part III-Question Answering(10 marks each question, 60 marks in total)

3-1

一个盒子中有 4 个红球, 3 个白球和 2 个黑球, 编号分别为红球 $1-4$ , 白球 $1-3$ , 黑球 $1-2$ . 现从中无放回地依次取出 3 个球, 求:

(1) 第三次才取到红球的概率；

(2) 已知第二次取到白球的条件下, 第一次取到黑球的概率；

(3) 三次取球中恰好有 1 个红球, 1 个白球, 1 个黑球.(顺序不限) 的概率

A box contains: 4 red balls(numbered 1-4) , 3 white balls(numbered 1-3) , 2 black balls (numbered 1-2) . Three balls are drawn without replacement in sequence. Find the following probabilities:

(1) The probability that the third ball is the first red ball drawn(i.e., the first two balls are not red, and the third is red)

(2) The conditional probability that the first ball is black, given that the second ball is white

(3) The probability that the three balls drawn consist of exactly one red, one white, and one black ball(in any order)

3-2

设 $E$ 和 $F$ 是某试验的样本空间中互不相容的事件, $E$发生概率为$P_1$, $F$ 发生概率为 $P_2$ . 假定重复做该试验至 $E$ 和 $F$ 有一个发生. 这个试验的样本空间是什么? 条件 $E$ 在事件 $F$ 之前发生的概率是多少?

Let $E$ and $F$ be mutually-exclusive events in the sample space of a certain experiment. The probability of $E$ occurring is $P_1$ , and the probability of $F$ occurring is $P_2$ . Suppose the experiment is repeated until either $E$ or $F$ occurs. What is the sample space of this experiment? What is the probability that event $E$ occurs before event $F$ ?

3-3

某抽奖活动规则如下:

• 抽奖箱中有 10 张券, 其中 3 张是一等奖, 4 张是二等奖, 3 张是谢谢参与.

• 玩家依次不放回地抽取 2 张券, $X$ 表示抽到一等奖的数量, $Y$ 表示抽到二等奖数量.

(1) 求 $(X,Y)$ 的联合概率分布表, 并验证其合法性；

(2) 分别求 $X$ 和 $Y$ 的边际分布, 并判断独立性；

(3) 求抽到一二等奖数量之和的随机变量 $Z$ 的概率分布.

A lottery event is conducted with the following rules:

• There are 10 tickets in a box: 3 are First Prize, 4 are Second Prize, and 3 are No Prize

• A player draws 2 tickets without replacement. Let $X$ be the number of First Prizes drawn, and $Y$ the number of Second Prizes drawn

Questions:

(1) Construct the joint probability distribution table for $(X,Y)$ and verify its validity

(2) Find the marginal distributions of $X$ and $Y$ , and determine if they are independent

(3) Define $Z=X+Y$ . Find the probability distribution of $Z$

3-4

设 $X\sim N(0,1)$

(1) 求 $Y=e^X$ 的概率密度函数

(2) 求 $Y=X^2+1$ 的概率密度函数

Let $X\sim N(0,1)$

(1) Find the probability density function of $Y=e^X$

(2) Find the probability density function of $Y=X^2+1$

3-5

假设随机变量 $X$ 和 $Y$ 独立, 且分别服从泊松分布 $X\sim P\left({\lambda }_1\right),Y\sim P\left({\lambda }_2\right)$

(1) 求 $X$ 和 $Y$ 的联合频率函数

(2) 求条件概率 $P(X=k\mid X+Y=n)$ , 其中 $n\ge k$ 是非负整数

Suppose random variables $X$ and $Y$ are independent and follow Poisson distributions $X\sim P\left({\lambda }_1\right)$ and $Y\sim P\left({\lambda }_2\right)$ , respectively

(1) Find the joint probability mass function of $X$ and $Y$ ；

(2) Find the conditional probability $P(X=k\mid X+Y=n)$ , where $n\ge k$ is a non-negative integer

$$\$\$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$$$\text{\#\#\# 3-6 令随机向量 \$(X, Y)\$ 的联合分布函数为}$$

F(x, y) = \begin{cases}c\left(1-e^{-2 x}\right) \left(1-e^{-3 y}\right) , & 0<x, 0<y \ 0, & \text { otherwise }\end

$$\$\${e^{-2x}|}_1^3$$

其中 $c$ 为常数

(1) 求出常数 $c$ 及 $X$ 和 $Y$ 的联合概率密度函数 $f$

(2) 求出 $X$ 与 $Y$ 的边缘概率密度函数 $f_X$ 和 $f_Y$ .试判断 $X$ 和 $Y$ 是否独立, 并说明理由

(3) 计算 $P(1<X<3,1<Y<2)$

Let $(X,Y)$ be a random vector with joint c.d.f

$$F(x,y)=\{\begin{array}{ll}c(1-e^{-2x})(1-e^{-3y}),& 0<x,0<y\\ 0,& \text{ otherwise }\end{array}$$

where $c$ is a constant

(1) Determine the value of $c$ and find the joint p.d.f.$f$ of $X$ and $Y$

(2) Find the two marginal p.d.f.s $f_X$ and $f_Y$ , and determine with reasons whether $X$ and $Y$ are independent

(3) Calculate $P(1<X<3,1<Y<2)$