Discrete random variable $\Theta$ is uniformly distributed between 1 and 100. Given $\Theta$ discrete random variable $X$ is uniformly distributed between 1 and $\Theta$. Show that
$E[X^2] = \frac{1}{100}\sum\limits_{x=1}^{100} x^2 \bigg(\sum\limits_{\theta=x}^{100} \frac{1}{\theta}\bigg)$
How to use $E[E[X^2|\Theta]]=E[X^2]$ to get the answer. Thank you
If $X$ and $Z$ are discrete, then $$ E(X) = E(E(X \mid Z)) \mathop{=}^{\mathrm{def}} \sum_z E(X \mid Z = z) P(Z = z). $$ If you know $E(X \mid Z = z) = u(z),$ then $$ E(X) = \sum_z u(z) P(Z = z). $$ Now apply this result replacing the appropriate values.