Square of gamma random variable If i have a random variable with distribution $X \sim \Gamma(\alpha,\beta)$ then what would be the distribution of $Y = \lambda X^2$ (with $\lambda$ a scaling factor)? Can I say that $Y$ will follow a gamma distribution as well?
 A: By definition, the probability element of $X$ has the form
$$f_X(x)dx \propto x^\alpha \exp(-x) \frac{dx}{x}$$
for $x\gt 0$ (and equal to $0$ otherwise).
Let $y = \lambda x^2$ (with $\lambda \gt 0$).  Because $x$ is positive, this is a one-to-one transformation, entailing both
$$x = \left(\frac{y}{\lambda}\right)^{1/2}$$
and
$$dy = 2\lambda x dx,$$
whence
$$\frac{dx}{x} = \frac{dy}{2\lambda x^2} = \frac{1}{2} \frac{dy}{y}.$$
Consequently the probability element for $Y$ is
$$f_Y(y) dy = \frac{1}{2}\left(\frac{y}{\lambda}\right)^{\alpha/2} \exp\left(-\left(\frac{y}{\lambda}\right)^{1/2}\right)\frac{dy}{y}.$$
By rescaling it we can recognize the PDF as having the basic form $y^{\alpha/2-1} \exp(-\sqrt{y})$.  It is a Generalized Gamma distribution.  Such distributions, by definition, are positive powers of Gamma distributions.
If $Y$ had a Gamma distribution, say with parameter $\beta$ and scale $\sigma$, then the ratio of the two (unnormalized) expressions for their PDFs would have to be some nonzero positive constant $C$.  Equivalently, for all $y\gt 0$ it must be the case that
$$y^{\alpha/2 - \beta} = C\exp\left(\sqrt{y/\lambda}-y/\sigma\right).$$
The right hand side decreases exponentially as $y\to\infty$ whereas the left hand side is only a power law: the two cannot be the same.
A: Assuming you're using the parametrization $f(x;\alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha - 1}e^{-\beta x}$, a result commonly seen in mathematical statistics classes is that
$$X \sim \Gamma(\alpha, \beta) \Rightarrow cX \sim \Gamma\left(\alpha, \frac{\beta}{c}\right).$$
After a ton of Googling it doesn't seem like there's a name for the distribution of a squared Gamma random variable.  However you can find its distribution using standard transformation of a random variable.
