Let $X ∼ Gamma(α, β)$, so that $X$ has pdf $$f(x)=\frac{β^α}{Γ(α)}x^{α-1}exp(−βx)$$ for $x > 0$ and $0$ otherwise
Find the pdf of the transformed variable: $Y = X^4$
Ive examined this question and I cant seem to find a route to the soluton so any help will be appreciated.
Call $F$ and $f$ the cdf and pdf of $X$ respectively. Call $G$ and $g$ the cdf and pdf of $Y$ respectively.
$$G(y)=P(Y\leq y) = P(X^4\leq y) = P(X\leq\sqrt[4]{y}) = F(\sqrt[4]{y}) $$
so that you have $$g(y) = \frac{dG(y)}{dt} =\frac{dF(\sqrt[4]{y})}{dt} = f(\sqrt[4]{y})\frac{y^{\frac{-3}{4}}}{4} = f(\sqrt[4]{y})\frac{1}{4\sqrt[4]{y^3}}$$
Hence you have $$g(y) = \frac{\beta^\alpha}{4\Gamma(\alpha)}y^{\frac{\alpha-1}{4}-\frac{3}{4}}e^{-\beta \sqrt[4]{y}} = \frac{\beta^\alpha}{4\Gamma(\alpha)}y^{\frac{\alpha}{4}-1}e^{-\beta \sqrt[4]{y}} $$