Calculate Expected value using conditioning

Question

How to calculate E[X] using conditioning. Let a random variable Y = X^2, While X is a random variable that can take the following values {-2,-1,0,1,2}. Each value can occur with equal probability i.e., 1/5.

Answer 1

We have \begin{align} \mathbb E[X] &= \sum_{n=0}^2 \mathbb E[X\mid Y=n]\mathbb P(Y=n)\\ &=\mathbb E[X\mid Y=0]\cdot\frac15 + \mathbb E[X\mid Y=1]\cdot\frac25 + \mathbb E[X\mid Y=4]\cdot\frac25\\ &=0+0+0=0. \end{align}

Answer 2

$$ E(X) = \left( \frac{1}{5} \right)(-2) + \left( \frac{1}{5} \right)(-1) + \left( \frac{1}{5} \right)(0) + \left( \frac{1}{5} \right)(1) + \left( \frac{1}{5} \right)(2) $$ $$ E(X) = \frac{ -2 -1 + 1 + 2} {5} $$ $$ E(X) = 0 $$ Now for the $E(Y)$. $$ E(Y) = \left( \frac{1}{5} \right)((-2)^2) + \left( \frac{1}{5} \right)((-1)^2) + \left( \frac{1}{5} \right)(0^2) + \left( \frac{1}{5} \right)(1^2) + \left( \frac{1}{5} \right)(2^2) $$ $$ E(Y) = \frac{ 4 + 1 + 0 + 1 + 4}{5} $$ $$ E(Y) = \frac{10}{5} = 2 $$