If X is a random continuous variable, what is the mean and variance of $X^2$? I'm not a math major and also using a non math-major book, and I can't understand the notation-heavy entry in wikipedia so go easy on me. =)
At first I wanted to go back to definition from the book for expected value and variance: $$E(X)= \int x f(x) dx$$ and $$V(X)=\int (x-\mu)^2 f(x) dx.$$
The alternative form $V(X)$ was given as $E(X^2) - E(X)^2$; from the derivation of the form, I noticed that $E(X^2)$ is$\int x^2 f(x) dx$ so is $V(X^2)=\int (x^2 -\mu)^2 f(x) dx$? (Question 1)
Also from $V(X)=E(X^2) - E(X)^2$, is $E(X^2)=\mu^2 + \sigma^ 2?$ (Question 2)
Is there a manageable expression for $V(X^2)$ in terms of $\sigma$ and $\mu$? (Question 3)
Since $V(X)=E(X^2) - E(X)^2$, plugging in $X^2$ in place of $X$ gets you $$ V(X^2)=E(X^4) - E(X^2)^2.\tag1 $$ You are right that $E(X^2)=\mu^2 + \sigma^ 2$ (question 2), but for question 1 you need to compute $E(X^4)$ using $$ E(X^4)=\int x^4 f(x)\,dx.\tag2 $$ As for question 3: Unless you have a specific distribution in mind, there is no way in general to get $E(X^4)$ from $\mu$ and $\sigma^2$ -- you need to work out the integral (2).