Lets say I have some function, say $f(x)$. I'll define a random variable $p$ with some distribution between some bounds $a$ & $b$. How can I find the distribution of $f(p)$?
For example. With $f(x)=x^2$ and $p$ a uniform distribution over $[-1,1]$ the distribution looks a follows:
Is there an analytic way to do these type of calculations?
If $X$ has uniform distribution on $(-1,1)$ and $f(x)=x^{2}$ then $P(f(X)\leq t)=P(|X|\leq \sqrt t)=\sqrt t$ for $0 <t<1$. Differentiating this we get the density of $f(X)$ as $\frac 1 2 t^{-1/2}$ for $0 <t<1$.