Let $p_1$ have $p_x(x) = 0.3*0.7^X$. Let $p_2$ have the exact same $p_x(x) = 0.3*0.7^X$. Assume this is valid for all $x\geq 0$.
Assume that $p_1,p_2$ are independent random variables.
Let $Y=(p_1)(p_2)$. Calculate $var(Y)$.
Here is my attempt:
Usually my process for this is to calculate $E[Y], E[Y^2]$, and then apply the formula (squaring $E[Y]$ as a whole, of course]
So, $E[Y]=E[p_1p_2]=E[p_1]E[p_2]$. Calculating one will give me the other, as they are the exact same.
I notice that $p_1$ is $Geo(\theta=0.3)=\theta(1-\theta)^X$. The expectation is given by $(1-\theta)/\theta=0.7/0.3=2.33$
This helps me with $E[Y]=(2.33)(2.33)$, but how do I calculate $E[Y^2]$?