I would like to understand what am I doing:
So for example, to find $\mathbb E[f(x)]$ I compute $\int x f(x) $
To compute $Var [f(x)]$ I compute $\int x^2 f(x) $, where does the $x^2$ come from?
When $X \sim N(0,5)$ I need $\mathbb E[X]= \int \sqrt 5 Z \times exp(-z^2/2) dz $
My questions:
1) For a random variable $X$, why don't I need to multiply by x just like in the first example? Why is it simply the integral of $X$ and it's probability distribution function, which in this case is just the standard pdf?
2) In the first example, does it mean that $x$ is the pdf?
3) How can one calculate variance using pdf? Let's say find the variance of $Y=e^{kX}$