Let \(X\sim\text{Gamma}(\alpha,\,\beta)\), and define a new random variable \(Y=X^r\) for any positive constant \(r>0\).
- What is the density of \(Y\)?
- Derive a formula for \(E(Y^n)\) that works for any \(n\in\mathbb{N}\).
Let \(X\sim\text{Gamma}(\alpha,\,\beta)\), and define a new random variable \(Y=X^r\) for any positive constant \(r>0\).