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Gamma distribution

The waiting time until the αth event in a Poisson process; a flexible positive, right-skewed severity model.

Parameters and support

$\alpha$

shape - number of events waited for - $\alpha > 0$

$\lambda$

rate (inverse scale) - $\lambda > 0$

Support

$x > 0$

The formulas

f(x)

$$f(x) = \dfrac{\lambda^{\alpha} x^{\alpha - 1} e^{-\lambda x}}{\Gamma(\alpha)}$$

F(x)

$$F(x) = \dfrac{\gamma(\alpha,\, \lambda x)}{\Gamma(\alpha)}$$

Elementary only when α is a positive integer, where it collapses to a finite Poisson sum: F(x) = 1 − Σ e^(−λx)(λx)^k/k! for k = 0…α−1.

Mean

$$E[X] = \dfrac{\alpha}{\lambda}$$

Variance

$$\operatorname{Var}(X) = \dfrac{\alpha}{\lambda^{2}}$$

MGF

$$M(t) = \left(\dfrac{\lambda}{\lambda - t}\right)^{\alpha}, \quad t < \lambda$$

Where the moments come from

1. Write $X = \sum_{k=1}^{\alpha} E_k$ with independent Exponential($\lambda$) waits (integer $\alpha$).
2. $E[X] = \alpha \cdot \frac{1}{\lambda}$ and, by independence, $\operatorname{Var}(X) = \alpha \cdot \frac{1}{\lambda^2}$.
3. The MGF is the exponential MGF to the $\alpha$th power, $\left(\frac{\lambda}{\lambda-t}\right)^{\alpha}$, and that form holds for ALL $\alpha > 0$.
4. Differentiating the MGF twice at 0 gives $E[X] = \alpha/\lambda$ and $E[X^2] = \frac{\alpha(\alpha+1)}{\lambda^2}$, confirming the variance.
5. Gamma($\alpha = 1$) is exponential; Gamma($\alpha = k/2$, $\lambda = 1/2$) is chi-square with $k$ degrees of freedom.

Worked example

The mean, variance, CDF and moment generating function above are re-derived numerically from this distribution’s own density on every test run - summed over the support for a discrete distribution, integrated by quadrature for a continuous one - and compared with the closed forms printed here. A typo on this page fails that test run.

Traps

• Mixing the rate parameterisation (mean α/λ) with the scale parameterisation (mean αθ). They are reciprocals.
• Trying to integrate the CDF for non-integer α - it has no elementary form.
• Forgetting that summing independent gammas requires the SAME rate λ.

Related

• Exponential distribution
• Poisson distribution
• Chi-square distribution
• Normal distribution

Drill this: the Exam P question bank has original questions on this distribution, and the recall trainer builds its prompts from exactly the formulas above.