课程名称︰高等统计学上
课程性质︰选修
课程教师︰蒋明晃
开课学院：管理学院
开课系所︰工管系
考试日期（年月日）︰20181107
考试时限（分钟）：9:10~12:10,180 mins
试题 :
1. A type C battery is in working condition with probability 0.7, whereas a
type D battery is in working condition with probability 0.4. A battery is
randomly chosen from a bin consisting of 8 type C and 6 type D batteries.
(a) What is the probability that the battery works? (b) Given that the
battery does not work, what is the conditional probability that it was type
C battery? (8%)
2. Two local factories, A and B, produce radios. Each radio produced at
factory A is defective with probability 0.05, whereas each one produced
at factory B is defective with probability 0.01. Suppose you purchase two
radios that were produced at the same factory, which is equally likely to
have been either factory A or factory B. If the first radio that you check
is defective, what is the conditional probability that the other one is also
defective? (8%)
3. Let U be a uniform (0,1) random variable, and let a<b be constants.
(a) Show that if b>0, then bU is uniformly distributed on (0,b) (4%)
(b) Show that a+U is uniformly distributed on (a, a+1) (4%)
(c) Show that min(U,1-U) is a uniform (0,1/2) random variable. (8%)
(d) Show that max(U,1-U) is a uniform (1/2,1) random variable. (8%)
4. The joint density of X and Y is given by
f(x,y)=x*e^(-x*(1+y)) for 0<x<inf, 0<y<inf
=0 otherwise
(a) Find the conditional density function of X,given Y=y, and that of Y,
given X=x. (8%)
(b) Find the density function of the random variable Z=XY. (10%)
5. A certain river floods every year. Suppose the low-water mark is set at 1
and the high-water mark Y has distribution function
F(y)=1-1/y^2, 1≦y≦inf
(a) Verify that F(y) is a cdf and find the pdf of Y. (5%)
(b) If the low-water mark is reset at 0 and we use a unit of measurement that
is 1/10 of that given previously, the high-water mark becomes Z=10(Y-1).
Find the F(z). (8%)
6. Let X has pdf f(x)=2(x+1)/9, -1<x<2
Compute the probability density function of Y=X^2 (10%)
7. A and B are two events. Prove the following statements:
(a) If A is the subset of B, the P(A|B)=P(A)/P(B)
(b) If A and B are independent and P(A)>0 and P(B)>0, they cannot mutually
exclusive.
(c) If A and B are mutually exclusive, then P(A|A∪B)=P(A)/(P(A)+P(B)) (9%)
8. An ambulance travels back and forth at a constant speed along a road of
length L. At a certain moment of time, an accident occurs at a point uniformly
distributed on the road. [That is, the distance of the point from one of the
fixed ends of the road is uniformly distributed over (0,L).] Assuming that
the ambulance's location at the moment of the accident is also uniformly
distributed, and assuming independence of the variables, derive the
distribution of the distance of the ambulance from the accident. (10%)