Download Homework 3

Homework 3
1. Let the joint density function of X 1 , X 2 , X 3 be
f ( x1 , x2 , x3 )  kx1 x22 x3 , 0  x1 , x2  1;0  x3  2
0, otherwise.
(a) Find k.
1
1


(b) Find P X 1  , X 2  ,1  X 3  2  .
4
2


(c) Find the joint probability density function of X 2 and X 3 .
(d) Find the marginal density function of X 2 .
2. The joint probability distribution table of the discrete random variables X
and Y is as follows.
Y
X
0
1
(a) Find P X  Y  3 .
(b) Find E X  and E Y .
(c) Find Var X  and Var Y .
1
2
3
1/4
1/6
1/6
1/8
1/6
1/8
(d) Find EY | X  0 and EY | X  1 .
3. Let the random variables X and Y be i.i.d. with the probability density
function f x  2x,0  x  1, and 0 otherwise. Find the conditional probability
P X  Y | X  2Y  .
4. Suppose the joint probability density function of X and Y is given by
f ( x, y )  4 y  x  y e   x  y  , 0  x  ;0  y  x
0, otherwise.
Compute EX | Y  y .
5. Let the conditional probability (or density) function of X given    is
f x |   and the marginal probability (or density) function  is g   .
Describe Bayes theorem for the cases that
(a) X and  being discrete; (b) X and  being continuous.
6. Find the maximum likelihood estimators of the following parameters:
independent
(a)
Xi
~


N xi ,  2 , i  1,, n , where xi are known and  ,  2
are parameters.
independent
(b)
Xi
~
Poisson xi , i  1,, n , where xi
are known and  is
the parameter.
7. Find the maximum likelihood estimators of  ,1 and  2 for the random
sample with sample size n from the uniform distribution:
i .i .d .
i .i .d .
i .i .d .
1
1

(a) X i ~ U 0,  . (b) X i ~ U   ,   . (c) X i ~ U 1 , 2  .
2
2

(d) X i ~ U 1  c, 2  c  , c is a known constant. (e) X i ~ U  ,   1 .
i .i .d .
i .i .d .
8. Find the maximum likelihood estimators of  and  2 for the random
sample with sample size n from the normal distribution:
(a) X i ~ N  ,  . (b) X i ~ N  ,1 ,  is an integer. (c) X i ~ N  ,1 ,   0 .
i .i .d .
i .i .d .
i .i .d .
(d) X i ~ N 1,  2 ,  2  c , where c is a known positive constant.
i .i .d .
X i ~ f x |   
i .i .d .
9. (a)
1  x 
e
,  x  , i  1,,2m  1 , m is a positive integer.
2
Find the M.L.E. of  .


(b) X i ~ N  ,2, i  1,, n . Find the M.L.E. of log  2    1 .
i .i .d .
(c) The probability distribution function of the i.i.d. random variables X and
Y is f 1   , f 2  0.4   , f 3  0.6  2 , f x  0, otherwise. Find the
M.L.E. of  .
X i ~ f x  
i .i .d .
(d)
1
 

x
 1
e
x

,0  x  , i  1, , n . Find the
estimating equations to obtain the maximum likelihood estimators of 
and  .
10.
Let
 X i  i.i.d .   X
  ~ N 
 Yi 
  Y
   X2
, 
   X  Y
 X  Y 
  Bivariate Normal , i  1,, n ,
 Y2 
where  X and Y are the means of X i and Yi , respectively,  X2 and
 Y2 are the variances of X i and Yi , and  is the correlation coefficient
of X i and Yi . Find the maximum likelihood estimators of  X , Y ,  X2 ,
 Y2 , and  .