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 $\theta > 0$:

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

$E[X] = \alpha\theta$, $\text{Var}(X) = \alpha\theta^2$. 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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