上外贤达学院概率论与数理统计第一学期(英文版)试卷

上外贤达学院概率论与数理统计第一学期(英文版)试卷

1 Let A,B be random events, and

A B,p(B) 0,then ( )

设A,B为随机事件，且A B,p(B) 0，则 （ ）

A p(A) p(A|B) B p(A) p(A|B) C p(A) p(A|B) D p(A) p(A|B)

2.Let A, B be two events, P(A) = 0.4，P(B) = 0.3，P( B) = 0.6，then P(AB) =( )

设A,B为随机事件, 且P(A) = 0.4，P(B) = 0.3，P( B) = 0.6，则 P(AB) =( ) A 0.18 B 0.6 C 0.3 D 0.4 3.Let the distribution function of the random variable

1

x2 A Be2,x 0 F(x)

x 0 0,

X be

A and B are equal to ( )

设随机变量X的分布函数为

1

x2 A Be2,x 0，则A和B等于 （ ） F(x)

x 0 0,

A A=1,B=1 B A=1,B=-1 C A=-1,B=1, D A=-1,B=-1 4.Suppose that random variables Xand Yare independent, if they have the same

1

probability distribution with 1/3

2

, thenP(X Y) ( ) 2/3

2

， 2/3

1

设随机变量X和Y相互独立，且具有相同的分布 1/3

则P(X Y) ( ) A 2/3

B 1

C 1/2

D 5/9

5.Suppose a random variable X has the probability distribution with

上外贤达学院概率论与数理统计第一学期(英文版)试卷

12 202

, then P(X 1)= ( )

0.30.10.20.4

12 202

设随机变量X具有概率分布 ，则P(X 1)= ( )

0.30.10.20.4 A 0.9 B 0.7 C 0.2 D 0.1

Part II (85%)

1. (10%)Suppose that four guests check their hats when they arrive at a restaurant, and these hats are returned to them in a random order when they leave. Determine the probability that no guest will receive the proper hat.

假设4位顾客去餐厅吃饭，帽子交与餐厅服务员保管，当他们离开时，服务员将帽子归还给他们，计算没有一位顾客拿回自己帽子的概率。

2. (10%) Suppose that a box contains one blue card and four red cards, which are labeled A,B,C and D. Suppose also that two of these five cards are selected at random ,without replacement. If it is known that at least one red card has been selected, what is the probability that both cards are red?

设一个盒子里有一个篮色的卡片和4个红色的卡片，四个红色的卡片分别标记为A,B,C,D。从中无放回的随机取出2张卡片。如果已知至少取到一个红色的卡片，试计算两张卡片都是红色的概率。

3.(15%)A machine produces defective parts with three different probabilities

depending on its state of repair. If the machine is in good working order,it produces defective parts with probability 0.02. If it is wearing down, it produces defective parts with probability 0.1. If it needs maintenance, it produces defective parts with probability 0.3. The probability that the machine is in good working order is 0.8, the probability that it is wearing down is 0 .1, and the probability that it needs maintenance is 0.1. Compute the probability that a randomly selected part will be defective.

一台机器生产不合格部件的概率与机器所处的状态有关。如果机器处于比较好的工作状态，则该概率为0.02；如果机器处于磨损状态，则该概率为0.1；如果机器需要维修，则该概率为0.3. 而机器处于比较好的工作状态的概率为0.8，处于磨损状态的概率为0.1，处于需要维修状态的概率为0.1.求随机选择一个部件，该部件为不合格品的概率。

4 (10%)Two students A and B are both registered for a certain course. Assume that student A attends class 80 percent of the time, student B attends class 60 percent of the time, and the absences of the two students are independent.

a What is the probability that at least one of the two students will be in class on a given day?

b If at least one of the two students is in class on a given day, what is the probability that A is in class that day?

两个学生A 和B 都选了同一门课。假定学生A 的出勤率为80%，学生B的出勤率为60%，两个学生是否出勤是相互独立的。求

上外贤达学院概率论与数理统计第一学期(英文版)试卷

（a）某一天至少有一个学生出勤的概率；

（b）假定某一天至少有一个学生出勤，求出勤的学生是A的概率。

5.(10%) Suppose that a random variable X has the binomial distribution with parameters n=15 and p=0.5. Find 设X

p(X 6).

~b(15,0.5)，求p(X 6)。

2x,for0 x 1f(x)

0,otherwise

6（15%） Suppose that X has the p.d.fFind the c.d.f of X. 设X的概率密度函数为

2x,for0 x 1f(x)

0,otherwise

，求X的分布函数。

7. (15%)Suppose that the p.d.f of a random variable X is as follows

3x2,for0 x 1f(x)

0,otherwise

13

(a) Determine the values of the probabilities p( X ).

44

(b) Also suppose that

Y 1 X2.Determine the p.d.f of Y.

，

3x2,for0 x 1

设随机变量X 的概率密度函数为f(x)

0,otherwise

(a)计算

132p( X )；（b）设Y 1 X，求Y的概率密度函数. 44

答案

Part I．Choice (3*5,15%)

上外贤达学院概率论与数理统计第一学期(英文版)试卷

Part II (85%)

Solution: Let

Ai denote

the i-th guest receive the proper hat(i=1,2,3,4), then

p(Ai)

Let

1

.(3 points) 4

B denote the event that no guest will receive the proper hat, then

B (A1

Thus

4

A2A3

A4)c.(5 points) A2

A3

A4)

p(B) 1 p(A1

1 [ p(Ai) p(AiAj)

i 1

i j

i j k

p(AiAjAk) p(A1A2A3A4)

1113

=1 [1 ]=(10 points).

2!3!4!8

2. Solution Let A denote the event that at least one red card has been selected and B denote the event that both cards are red;(3points)

p(AB)p(B)

So p(B|A) (5points)

p(A)p(A)

212

C4C4 C43

p(B) 2 (7points),p(A) 1(9points) 2

C55C5

3

p(B|A) (10points).

5

3. Solution Let Let Let Let

A1 denote the event that the machine is in good working order. A2 denote the event that the machine is wearing down. A3 denote the event that the machine needs maintenance.

B denote the event that the part selected is …… 此处隐藏：3609字，全部文档内容请下载后查看。喜欢就下载吧 ……