1 Let A,B be random events, and A?设A,B为随机事件，且A? 2.Let A, B be two events, P(A) = 0.4，P(B) = 0.3，P(A?B) = 0.6，then P(AB) =( )
C p(A)?p(A|B) D p(A)?p(A|B) A p(A)?p(A|B) B p(A)?p(A|B)
B,p(B)?0,then ( )
B,p(B)?0，则 （ ）
设A,B为随机事件, 且P(A) = 0.4，P(B) = 0.3，P(A?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 X be
F(x)????A?Be?12x2,x?0??0,x?0A and B are equal to
( ) 设随机变量X的分布函数为
F(x)???1?A?Be?2x2,x?0，则A和B等于 （ ）??0,x?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
probability distribution with??12???1/32/3???, thenP(X?Y)? ( )
设随机变量X和Y相互独立，且具有相同的分布??12???1/32/3??，?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
???2012??0.30.10.20.4?2?, then P(X?1)= ( ) )
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
得分
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（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。
设X答案
题号
答题区
Find the c.d.f of X.
(b) Also suppose that Yparameters n=15 and p=0.5. Find
?3x2,for0?x?1f(x)???0,otherwise~b(15,0.5)，求p(X?6)。
?2x,for0?x?1设X的概率密度函数为f(x)???0,otherwise13(a) Determine the values of the probabilities p(?X?).
447. (15%)Suppose that the p.d.f of a random variable X is as follows
?3x2,for0?x?1设随机变量X 的概率密度函数为f(x)???0,otherwise?2x,for0?x?16（15%） Suppose that X has the p.d.ff(x)???0,otherwise132(a)计算p(?X?)；（b）设Y?1?X，求Y的概率密度函数.
445.(10%) Suppose that a random variable X has the binomial distribution with
Solution:
1
B
Part II (85%)
2
Part I．Choice (3*5,15%)
?1?X2.Determine the p.d.f of Y.
C
3
B
4
p(X?6).
D
5
A
得分
，求X的分布函数。
，
Let Thus
4i?1
Let
31113=1?[1???]=(10 points).
2!3!4!83. Solution
1p(Ai)?.(3 points)
4?1?[?p(Ai)??p(AiAj)?B?(A1?A2?A3?A4)c.(5 points)
3p(B|A)?(10points).
5Ai denote the i-th guest
So according to the law of total probability
p(AB)p(B)?So p(B|A)?(5points)
p(A)p(A)p(B)??p(Ai)p(B|Ai) =0.056(15points).
i?jp(B)?1?p(A1?A2?A3?A4)
Let A2 denote the event that the machine is wearing down.
212C4C4?C43p(B)?2?(7points),p(A)??1(9points)2C55C5Let A3 denote the event that the machine needs maintenance.
Let B denote the event that the part selected is defective.(4points)
Then p(A1)?0.8,p(A2)?0.1,p(A3)?0.1,
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(B|A1)?0.02,p(B|A2)?0.1,p(B|A3)?0.3(10points)
Let A1 denote the event that the machine is in good working order.
B denote the event that no guest will receive the proper hat, then
i?1receive the proper hat(i=1,2,3,4), then
i?j?k?p(AiAjAk)?p(A1?A2?A3?A4)
(1)If y(10points)
5k?07. Solution
Thus,
k15155. Solution
(10points).
6. Solution If xIf 0?x?1,F(x)??x???0,x?0?2Thus F(x)??x,0?x?1.(15points).
?1,x?1?p(X?6)??C(0.5)(4points)??eIf x?1,F(x)?1.(12points)
?0,F(x)?0. (4points)
25k?0(2)Ify?1,FY(y)?p(Y?y)?p(1?X?y)?7.5(b)FY(y)?p(Y?y)?p(1?X?y)(6points)
20x3313252344 (a)p(?X?)??13xdx?x|1??0.39(5points)
4464444. Solution Let A denote the event that student A attend the class.
Let B denote the event that student B attend the class.
p(A(A?B))p(A)??0.869(15points). (b)p(A|A?B)?p(A?B)0.92 (a) p(A?B)?p(A)?p(B)?p(AB)?0.8?0.6?0.8?0.6?0.92?p(1?y?X2)?p(X?1?y)?p(X??1?y)Then p(A)?0.8,p(B)?0.6(5points)
?1?p(X?1?y)?p(X??1?y)(12points)
?1,FY(y)?p(Y?y)?p(1?X2?y)?1(8points)
f(t)dt??2tdt?x2.(8points)
7.5k(8points)?0.24k!??31?y,0?y?1So fY(y)??(15points).
??0otherwise
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