How do you determine the mode of a gamma distribution with parameters $\alpha$ and $\beta$ ? Without looking on Wikipedia.
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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.
angryavian
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How do we know it is zero when $x=\frac{\alpha-1}{\beta}$ or $x=0$. Zero is not the mode because $x>0$ correct? – Amanda R. Mar 30 '17 at 06:58
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Look at the curve : It is not 0 because in 0, the pdf is 0 ! – Jean Marie Mar 30 '17 at 07:04
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@JeanMarie can you please explain more – Amanda R. Mar 30 '17 at 07:08
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You are looking for a mode: it is necessarily a values of $x$ such that the pdf f(x) has a $>0$ value. In $0$ it has a zero value at least for the cases $\alpha>1$ (for the other cases, one cannat speak about a mode because the pdf has a vertical asymptote. – Jean Marie Mar 30 '17 at 07:22
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@JeanMarie Thanks for the clarifications! – angryavian Mar 30 '17 at 18:43