Determine the mode of the gamma distribution with parameters $\alpha$ and $\beta$
By Michael Henderson •
How do you determine the mode of a gamma distribution with parameters $\alpha$ and $\beta$ ? Without looking on Wikipedia.
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$\begingroup$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.
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