Calculate $\mathbb{E}(X^2)$ given other values

Question

Given the continuous random variable $X$ and the probability density function $$ f(x) = x -\frac{1}{4}x^3;\;\; 0\leq x \leq 2 $$ I have worked out the median and mean $\mathbb{E}(X)$ to be $1.08$ and $16/15$ respectively. How do i calculate $\mathbb{E}\left(X^2\right)$ and then the variance $\sigma^2=\mathrm{Var}(X)$ and the standard deviation $\sigma$.

Answer 1

$$ \mathbb{E}(g(X)) = \int_{a}^bg(X)f_X(X)dx $$ so we can solve with $g(x) = x^2$. Then you have the variance as $$ \mathrm{Var}(X) = \mathbb{E}(X^2) - \mathbb{E}^2(X) $$ The standard deviation is trivial after the variance.