Download doc

機 率 Probability
(2007 Spring - Final)
1. 請按順序作答(兩題一頁或一題一頁)
2. 答題須寫下式子,再計算數字答案(或查表求得),
不能直接僅寫數字答案(否則將不予給分*).
3. 期末考成績, 總成績, 與補考名單 6/30(星期六) 中午前公布於 course page
Time: 13:30-15:30 6/28/2007
Place: EC016
9 problems and 100 points in total
1. Let X be uniformly distributed over (0, 1) and let Y = X n , where n is a positive
integer.
(a) Find the distribution function of Y.
(b) Find the density function of Y. (8 points)
2. If X is a normal random variable with parameters   3 and  2  16 , find
(a) P( 1 < X < 11 );
(b) P( X > 0);
(c) P( | X – 3 | > 6 ). (6 points)
(Use the attached normal distribution table.)
3. Let Z be the standard normal random variable Z. And a, b, c, d are the positive
values such that
P( -a < Z < a ) = 0.50,
P( -b < Z < b ) = 0.90,
P( -c < Z < c ) = 0.95,
P( -d < Z < d ) = 0.99.
What are a, b, c, and d? (8 points) (a, b, c, d 允許有 error 0.01)
(Use the attached normal distribution table.)
4. (a) What is CLT (Central limit theorem) ? (5 points)
(b) A fair die is rolled 20 times. What is the approximate probability that
the sum of the outcomes is between 65 and 75? (7 points)
(Use the attached normal distribution table.)
5. Let X and Y be two independent random variables with the same density function
given by
e  x
f ( x)  
0
0 x
o t h e r w. i s e
(a) Find P(X < 3Y). (6 points)
(b) Find P(min(X, Y) > 1).(3 points)
(c) Find P(max(X, Y) > 1). (3 points)
1
6. First, a point Y is selected at random from the interval (0, 1). The another point
X is chosen at random from the interval (0, Y).
(a) Find Var(Y2). (3 points)
(b) Find conditional expectation E(X 2 | Y  y). (3 points)
(c) Find marginal density f X (x). (3 points)
(d) Find EX. (3 points)
7. Let X and Y be two independent random variables with expected values  X  1
and Y  2 , and variances  X2  3 and  Y2  4 , respectively.
(a) Calculate Var(2X-3Y+1) (3 points)
(b) Calculate  (2 X  3Y  1,2 X  3Y  1). (3 points)
(c) Calculate Var(XY). (3 points)
(d) Calculate Cov(X, XY).(3 points) (計算前務必列公式,否則不計分)
8. A group of N people throw their hats into the center of a room. The hats are
mixed up, and each person randomly selected one. Let random variable X
denote the number of people that select their own hats. We may write
1 if the ith man selected his own hat
X = X1 + X2 + … + XN, where X i  
0 otherwise.
(a) Find EXi. (2 points)
(b)
(c)
(d)
(e)
(f)
Find EX. (2 points)
Find Var(Xi). (3 points)
Find E(XiXj) for different i, j. (3 points)
Find Cov(Xi, Xj) for different i, j. (3 points)
Find Var(X) by using (b) and (c). (3 points)
9. (a) (Markov’s inequality) Let X be a nonnegative random variable.
Prove that for any t > 0, P( X  t ) 
EX
. (6 points)
t
(b) Let X be a random variable with EX=0 and Var(X)=  2 . (8 points)
Use (a) to prove that for any t > 0, P( X  t ) 
2
 2  t2
.
(Hint: similar to the proof of Chebyshev inequality)
2