Random Variable, Density Functions, Variance

Question

I know this question has been solved, but none of the online forums really explain it well, how are the values supposed to be placed into the formulas/theorems.

Let $X$ be a random variable with range $[-1, 1]$ and let $f_{X}(x)$ be the density of function $X$. Find $\mu(X)$ and $\sigma^{2} (X)$ if, for $\def\abs#1{\left\lvert{#1}\right\rvert}\abs x < 1$,

1. $f_X(x) = 1/2$
2. $f_X(x) = \abs x$
3. $f_X(x) = 1 − \abs x$
4. $f_X(x) = (3/2)x^{2}$

Thank you.

Answer 1

You may use the usual definitions. For pdf whose support is $[-1;1]$ then: $$\begin{split}\mu(X)&=\int_{-1}^1 x~f_X(x)~\mathsf d x\\[2ex]\sigma^2(X) &= \int_{-1}^1 (x-\mu(X))^2~f_X(x)~\mathsf d x\end{split}$$

However, @Bungo suggests an excellent shortcut, by noting the symmetry in each of the probability density functions.