Ask property of gamma distribution

Question

If $\Sigma X_i \sim gamma(2,\lambda)$, then $\Sigma X_i/\lambda \sim gamma(2,1)$. Why?

The background of this question is from George Casella textbook Example 9.2.3.

Answer 1

Let $X \sim \text{Gamma}(\alpha, \beta)$ with parameters shape and scale. Consider $Y=aX$ for some positive number $a$. By the transformation theorem we have that the pdf of $Y$ is $\frac{(1/\beta)^\alpha}{\Gamma(\alpha)}\left(\frac y a\right)^{\alpha-1}e^{-\frac 1 \beta \frac y a}\frac 1 a$ for $0<y<\infty$ which can be recognized as the pdf of a $\text{Gamma}(\alpha ,a\beta)$ distribution.

Directly applying this to your problem yields $\sum X_i\sim \text{Gamma}(2, \lambda)$ implies $\sum X_i/\lambda \sim \text{Gamma}(2, 1)$.