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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)

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

F(x)

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

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

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

Variance

$$\mathrm{Var}(X)=\frac{\alpha }{{\lambda }^2}$$

MGF

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

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, $\mathrm{Var}(X)=\alpha \cdot \frac{1}{{\lambda }^2}$.
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 the build.

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.