The Gamma Distribution and the Poisson Connection

The gamma density looks intimidating with its $\Gamma (\alpha )$, but the exam rarely makes you integrate it. For integer shape, a gamma probability is just a Poisson sum, the same $e^{-\lambda }{\lambda }^k/k!$ terms you already know.

$X\sim \text{Gamma}(\alpha ,\theta )$ with shape $\alpha >0$ and scale $\theta >0$:

$$f(x)=\frac{x^{\alpha -1}{e}^{-x/\theta }}{{\theta }^{\alpha }\Gamma (\alpha )},x>0$$

$E[X]=\alpha \theta$, $\text{Var}(X)=\alpha {\theta }^2$. You rarely integrate this. Instead, read the parameters off the form: the power on $x$ is $\alpha -1$, and the exponent gives $1/\theta$.

When $\alpha =1$, the gamma is the exponential Exp($\theta$). More generally, for integer $n$, a Gamma($n,\theta$) is the sum of $n$ independent Exp($\theta$) variables. That is why claim-waiting-time problems, which add up exponential gaps, produce gammas.

A Gamma($\alpha ,\theta$) with integer $\alpha$ is the waiting time for the $\alpha$-th event in a Poisson process of rate $1/\theta$. So a gamma tail is a Poisson sum:

Common mistakes

• Integrating the gamma density by hand. For integer $\alpha$, the Poisson connection replaces the integral with a finite sum. Reach for it before integrating.
• Flipping the inequality in the Poisson connection. $P(X\le t)=P(\text{Poisson}\ge \alpha )$: the continuous lower tail maps to the discrete UPPER tail. The waiting time is short exactly when many events have already occurred.
• Confusing scale and rate. SOA uses scale $\theta$ with $E=\alpha \theta$. A rate parametrization uses $\beta =1/\theta$ with $E=\alpha /\beta$. Read the density's exponent to get $\theta$.

Bottom line

• Gamma($\alpha ,\theta$): shape $\alpha$, scale $\theta$. $E[X]=\alpha \theta$, $\text{Var}(X)=\alpha {\theta }^2$.
• Read $\alpha ,\theta$ off the density: the power of $x$ is $\alpha -1$; the exponent is $-x/\theta$.
• $\alpha =1$ is the exponential; a Gamma($n,\theta$) is the sum of $n$ independent Exp($\theta$).
• Poisson connection (integer $\alpha$): $P(X\le t)=P(\text{Poisson}(t/\theta )\ge \alpha )$; equivalently $P(X>t)=P(\text{Poisson}(t/\theta )\le \alpha -1)$.

Exam shortcut

Read $\alpha$ and $\theta$ straight off the density (power on $x$ is $\alpha -1$, exponent is $-x/\theta$), then $E=\alpha \theta$ and $\text{Var}=\alpha {\theta }^2$. For a probability with integer $\alpha$, convert to a Poisson($t/\theta$) sum. "Gamma tail is a Poisson sum." Lower tail of $X$ maps to the upper tail of the Poisson, and vice versa. "Power plus one is the shape."

The full lesson (about 1,683 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b
• 2c
• 2d
• 2e
• 2f

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