Page 75 - DMTH404_STATISTICS

Unit 5: Functions of Random Variables

5.4 Keywords Notes

Double exponential distribution: The distribution with the density function given by * is known
as double exponential distribution.

Range: It is the difference between the largest and the smallest observations.
5.5 Self Assessment

Fill in the blanks:
1. Let (X,Y) be a random vector with the joint ................. fx, y (x,y). Let g(x,y) be a real-valued
function defined on R x R. For z E 2, define
D = {(x, Y) : g(x, y)  z}
z
2. The distribution with the density function given by * is known as .................
3. An important application of the ................. approach is to determine distribution of the
sum of two independent random variables not necessarily identically distributed.
4. The ................. of the transformation is equal to unity.
5. ................. is the difference between the largest and the smallest observations.

5.6 Review Questions

1. Suppose X and Y are independent random variables, each having uniform distribution on
(0, 1). Determine the density function of Z = X + Y.
2. Suppose (X,Y) has the joint probability density function

f (x,y) = x + y, if 0 < x, y < 1
= 0, otherwise
Find the density function of Z = XY.
3. Suppose X and Y are independent r. vs with density function f(x) and distribution function
F(x). Find the density function of Z = min (X,Y).
4. Suppose X and X are independent random variables with gamma densities f (x ) given by
1 2 i i
1  i 1  i x
f (X ) = x i e , 0 < x < 
i i (   i ) i
= 0 otherwise

X
for i = 1, 2. Let Z = X + X and Z = 1 . Show that Z and Z are independent random
1 1 2 2 X  X 1 2
1 2
variables. Find the distribution functions of Z and Z .
2 1
5. (Box - Muller transformation) Let X and X be independent random variables uniformly
1 2
distributed on [0, 1]. Define
Z = (-2 log X ) cos (2 X ),
1/2
1 1 2
1/2
Z = (-2 log X ) sin (2 X )
2 1 2
Show that Z and Z are independent standard normal random variables.
1 2

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