Gamma Distribution

Compact study note.

Summary

The gamma distribution models positive continuous quantities, especially sums of exponential waiting times. This note uses shape-scale parameters.^[1]

Prerequisites

• Gamma Function
• Exponential Distribution

Definition

$X\sim \mathrm{Gamma}(k,\theta )$ with shape $k$ and scale $\theta$ .

Notation and Assumptions

$k>0$ and $\theta >0$ . The rate form uses $\lambda =1/\theta$ and must not be mixed with this scale form.

Parameters

$k>0$ ; $\theta >0$ .

Support

$(0,\mathrm{\infty })$ .

PMF or PDF

$f_X(x)=x^{k-1}{e}^{-x/\theta }/(\mathrm{\Gamma }(k){\theta }^k)$ for $x>0$ .

CDF

The CDF uses the lower incomplete gamma function: $F_X(x)=\gamma (k,x/\theta )/\mathrm{\Gamma }(k)$ .

Moments

$E[X]=k\theta$ , $\mathrm{Var}(X)=k{\theta }^2$ , and $M_X(t)=(1-\theta t{)}^{-k}$ for $t<1/\theta$ .

Essential Result

If $k$ is a positive integer, $X$ is the sum of $k$ IID exponential variables with common scale $\theta$ .

Small Example

For $k=3$ and $\theta =2$ , $E[X]=6$ and $\mathrm{Var}(X)=12$ .

Common Mistakes

• Writing the MGF with the rate formula while calling the parameter a scale.
• Forgetting the support is positive.

Connections

• Gamma Function
• Exponential Distribution
• Chi-Square Distribution

References