Page 76 - DMTH404_STATISTICS

Statistics

Notes 6. Suppose (X , X ) have the joint density function
1 2
f (x , x ) = 4x x if 0 <x , x < 1
1 2 1 2 1 2
= 0 otherwise.

X
Define Z = 1 and Z = X X . Determine the joint density function of (Z , Z )
1 X 2 1 2 1 2
2
7. Suppose X , ....., X are n independent random variables with the same distribution
1 n
2
N(,  ). Define
1 n
X   X i
n i 1

X is called the sample mean. Extending the m.g.f. approach for more than two random
  2 
variables, show that X has the distribution N  ,  .

 n 
Answers: Self Assessment

1. density function 2. double exponential distribution 3. transformation
4. Jacobian 5. Range

5.7 Further Readings

Books Introductory Probability and Statistical Applications by P.L. Meyer
Introduction to Mathematical Statistics by Hogg and Craig
Fundamentals of Mathematical Statistics by S.C. Gupta and V.K. Kapoor

68 LOVELY PROFESSIONAL UNIVERSITY