Problem 1 X is a random variable whose PMF is given by the f

Problem 1. X is a random variable whose PMF is given by the following Px(a) = { 1/7, if a = 0,plus or minus 1,plus or minus 2, or plus or minus 3, 0, otherwise. (a) Find the PMF of Y=X^2+2. (b) Find the mean value of Y using the following two expressions and verify that they are equal: E[Y]= Sigma infinite (x^2+2)Px(x), and E[Y] =Sigma y yPY(y).

Solution

y is calculated as x^2+2

The above table gives the pdf of x and y with extra rows for calcultating mean and variance .

Mean of x =0

Var of X = E(X^2)-0 = 4

Mean of Y as per table =6

Mean of Y as per E(x^2+2) = E(x^2)+E(2)

=4+2 =6

Hence both are the same.

verified

x^2p	0	2/7	8/7	18/7	4
xp	0	0	0	0	0
p	1/7	1/7,1/7	1/7,1/7	1/7,1/7	1
x	0	1,-1	2,-2	3,-3
y	2	3	6	11
p(y)	1/7	2/7	2/7	2/7	1
yp	2/7	6/7	12/7	22/7	6