Expected Value Using Integration

Continuous random variables live or die by their pdf. Computing expectation, variance, and moments is integration in a thin disguise.

Let $X$ be a continuous random variable with probability density function

$f(x)=2x,0\le x\le 1,$

and $f(x)=0$ elsewhere.

Compute:

(a) The expected value $E[X]$.

(b) The second moment $E[X^2]$.

(c) The variance $\text{Var}(X)=E[X^2]-(E[X]{)}^2$.