I'm a student and I'm studying random variables and very new to it. I was studying the Uniform distribution and in it, it calculates the Expected of $X^2$ by $$ E\left(X^2\right) = \int_{- \infty}^\infty x^2 p(x) dx $$ I understand the calculation of it, but if the Probability density function plot of $X$ is like that (I fully understand that) I can't imagine the Probability density function of the $X^2$ because if we span the Probability density function of $X$ which is within $[a,b]$ to $[a^2,b^2]$ then the integral of $\int_{a^2}^{b^2} p(\sqrt x) dx$ may not be equal to 1 I guess.

Could you help me figure out what is the Probability density function of $X^2$?

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    $\begingroup$ I actually computed the pdf of $X^2$ for a general random variable here: math.stackexchange.com/a/3043502/90543 $\endgroup$ – jgon Dec 23 '18 at 17:21
  • $\begingroup$ @jgon thank you. it was hard for me to understand your answer because I'm very new to it, but finally, I get it. so I was right, it is not that simple that in the basic tutorials they just integral $\int_{- \infty}^\infty x^2 p(x) d_x $ to find the Expected. nice answer. $\endgroup$ – Peyman mohseni kiasari Dec 23 '18 at 17:36

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