15.6 - Gamma Properties

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

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

The mean of a gamma random variable is:

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

Proof

The proof is left for you as an exercise.

Theorem

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.

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