15.6 - Gamma Properties

Here, after formally defining the gamma distribution (we haven't done that yet?!), we present and prove (well, sort of!) three key properties of the gamma distribution.

Gamma Distribution

A continuous random variable \(X\) follows a gamma distribution with parameters \(\theta>0\) and \(\alpha>0\) if its probability density function is:

\(f(x)=\dfrac{1}{\Gamma(\alpha)\theta^\alpha} x^{\alpha-1} e^{-x/\theta}\)

for \(x>0\).

Before we get to the three theorems and proofs, two notes:

  1. We consider \(\alpha>0\) a positive integer if the derivation of the p.d.f. is motivated by waiting times until α events. But the p.d.f. is actually a valid p.d.f. for any \(\alpha>0\) (since \(\Gamma(\alpha)\) is defined for all positive \(\alpha\)).

  2. The gamma p.d.f. reaffirms that the exponential distribution is just a special case of the gamma distribution. That is, when you put \(\alpha=1\) into the gamma p.d.f., you get the exponential p.d.f.

Theorem Section

The moment generating function of a gamma random variable is:

\(M(t)=\dfrac{1}{(1-\theta t)^\alpha}\)

for \(t<\frac{1}{\theta}\).

Proof

By definition, the moment generating function \(M(t)\) of a gamma random variable is:

\(M(t)=E(e^{tX})=\int_0^\infty \dfrac{1}{\Gamma(\alpha)\theta^\alpha}e^{-x/\theta} x^{\alpha-1} e^{tx}dx\)

Collecting like terms, we get:

\(M(t)=E(e^{tX})=\int_0^\infty \dfrac{1}{\Gamma(\alpha)\theta^\alpha}e^{-x\left(\frac{1}{\theta}-t\right)} x^{\alpha-1} dx\)

Now, let's use the change of variable technique with:

\(y=x\left(\dfrac{1}{\theta}-t\right)\)

Rearranging, we get:

\(x=\dfrac{\theta}{1-\theta t}y\) and therefore \(dx=\dfrac{\theta}{1-\theta t}dy\)

Now, making the substitutions for \(x\) and \(dx\) into our integral, we get:

Theorem Section

The mean of a gamma random variable is:

\(\mu=E(X)=\alpha \theta\)

Proof

The proof is left for you as an exercise.

Theorem Section

The variance of a gamma random variable is:

\(\sigma^2=Var(X)=\alpha \theta^2\)

Proof

This proof is also left for you as an exercise.