Question

A Suppose X and Y are random variables that only take on the values 0, 1, and 2. (That is, for both of their probability mass functions, p(z) = 0 for xメ0.1.2.) Suppose E(X-Ely and E X2 EY]. Prove that EEY (Write your answer in complete sentences, as this requires a proof.)

Answer 1

Let p(x) and q(y) be the pmf of x and y respectively.

_(I)

$E[X^{2}] = E[Y^{2}] \Rightarrow \sum_{0}^{2}x^{2}p(x) = \sum_{0}^{2}y^{2}q(y)$
_(II)
Multiplying equation (I) by 2 we get
_(III)
Subtracting equation (III) from equation (I) we get

Subtracting equation (I) from equation (II) we get


Since $p(0) and q(0) = 0 \forall x,y\neq 0,1,2$
and   $p(1) = q(1) , p(2) = q(2)\Rightarrow p(0) = q(0)$
Since the distribution of p(x) and q(y) are equal:
E[X³] = E[Y³]