Calculate $E(X^2)$ with X uniformly distributed on [-1,2]

Question

X is uniformly distributed on $[-1,2]$.
Hence the density function should be $\frac13$ and E(X) = $0.5$.
Now I want to calculate $E[X²]$. But to do that I need the density function of $X²$.
This density function should be $0$ except between $0$ and $4$.
Is $E[X²]$ uniformly distributed on $[0,4]$?

Answer 1

Guide:

You do not need the densitfy function of $X^2$. Suppose $f$ is the densify function of $X$.

To compute the expected value of $g(X)$, you can compute $\int_{-1}^2 g(x) f(x) \, dx$.

We just have to compute $\int_{-1}^2 x^2 f(x) \, dx$.

Also, $X^2$ is not uniformly distributed.