Statistics （I） Final Exam
Jan. 17, 2002
10 points for each problem.
（1）Using moment-generating functions , show that as    the gamma distribution ,
properly standardized , tends to the standard normal distribution.
（2）
【a】Let X 1 ,…,X n be a sample from an N（  x ,  2 ）distribution and Y 1 ,…,Y m be
an independent sample from an N（  y ,  2 ）distribution. Show how to use the F
2
S
1 n
2
( X i  X )2 and
distribution to find P（ x 2 >c）,where S x 

n  1 i 1
Sy
Sy 
2
1 m
 ( Y j  Y )2 .
m  1 j 1
2
【b】Find the approximate variance of
Sx
by Taylor series expansion.
2
Sy
（3）Suppose that X 1 ,…,X n is an i.i.d. sample of size n follows a geometric distribution
P（X = k）= p（1-p） k 1 .
【a】Find the method of moments estimate of p.
【b】Find the sampling distribution of the moment estimate derived in 【a】.
（4）Assuming that X~ N（ 0 ,  2 ）, find the moment generating function of X and show the
even moments of X are  2 n 
( 2n )!  2 n
.
2 n ( n! )
（5）Assume the joint density of X and Y is f(x,y) = 2 , 0  x  y .
【a】Find the best linear predictor（denoted by Ŷ  a  bX ）and the minimum mean
square predictor of Y in terms of X.
【b】Compare the mean square prediction errors of the two predictors derived from【a】.
（6）Let T be an exponential random variable with parameter  , and conditional on T. Let
U be uniform on [0,T].
【a】Find the unconditional mean and variance of U.
【b】 Find the correlation of T and U.
（7）Assume U i , i  1,2 , …,n are independent uniform random variables.
【a】Find the density of U ( n )  U ( 1 ) , where U ( n ) and U ( 1 ) are the maximum and
minimum of U i ' s , respectively.
【b】Find E( U( n )  U( 1 ) ).