Power (Square) of Different Distributions? If there is a PDF $\text{f}(x)$, and $x$ follows different distributions, for example, Rayleigh, Rice, Nakagami, and Weibull. Then what will be the distribution of $|x|^2$ for these distributions? Please refer me to any book, research paper or any article which helps me with this.
 A: In general, if $X$ has cdf $F(x) = P(X \leq x)$ and density function $f(x)$, the cdf for $X^2$ will be $P(X^2 \leq x) = P(-\sqrt{x} \leq X \leq \sqrt{x}) = F(\sqrt{x}) - F(-\sqrt{x}).$ Then assuming $F$ is differentiable, the density function for $X^2$ is $$\frac{d}{dx}F(\sqrt{x}) - F(-\sqrt{x}) = f(\sqrt{x}) / (2 \sqrt{x}) + f(-\sqrt{x}) / (2 \sqrt{x}).$$ See also this question and the very detailed answer given there.
A: You have some random variable $X$ with density function $f$ and CDF $F$, and you want to know something about the density and/or CDF of the random variable $Y := X^2$. You can do that by considering first the CDF of $Y$. 
Let $a$ be a fixed nonnegative number. Then
\begin{align*}
  \mathbb P(Y \leq a) &= \mathbb P(X^2 \leq a) \\
&= \mathbb P(-\sqrt a \leq X \leq \sqrt a) \\
&= F(\sqrt a) - F(-\sqrt a)
\end{align*}
which gives an explicit formula for the CDF of $Y$ in terms of that of $X$. Since the density is the derivative of the CDF, we can leverage this to get the density function of $Y$, which we'll call $g$:
\begin{align*}
  g(a) &= \frac{\mathrm d}{\mathrm da} \mathbb P(Y \leq a) \\
&= \frac{\textrm d}{\textrm d a} \left[ F(\sqrt a) - F(-\sqrt a) \right] \\
&= f(\sqrt a) \cdot \frac{1}{2 \sqrt a} - f(-\sqrt a) \cdot \frac{-1}{2 \sqrt a} \\
&= \frac{f(\sqrt a) + f(-\sqrt a)}{2 \sqrt a} \\
\end{align*}
and in particular if $f$ is symmetric, then we have
$$g(a) = \frac{f(\sqrt a)}{\sqrt a}.$$
