Determine the mode of the gamma distribution with parameters $\alpha$ and $\beta$

Question

How do you determine the mode of a gamma distribution with parameters $\alpha$ and $\beta$ ? Without looking on Wikipedia.

Answer 1

Hint: you want to maximize $x^{\alpha-1} e^{-\beta x}$ over $x \in (0,\infty)$.

The derivative is $e^{-\beta x}[(\alpha-1)x^{\alpha-2} - \beta x^{\alpha-1}] = x^{\alpha-2} e^{-\beta x} (\alpha-1-\beta x)$, which is zero when $x= \frac{\alpha-1}{\beta}$ or $x=0$.

• If $\alpha \ge 1$, direct inspection shows that $x=0$ is not the mode, since the pdf is zero there. Thus the other critical point $\frac{\alpha-1}{\beta}$ must be the mode.
• If $\alpha < 1$ the pdf has a positive asymptote at $x=0$. Moreover the derivative is strictly negative for all $x>0$, so it decreases from $\infty$ to $0$ as $x$ goes from $0$ to $\infty$.

Thanks to JeanMarie for the clarifications.