Question

State and prove additive property of gamma distribution.

Answer 1

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

1) We consider α > 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 α > 0 (since Γ(α) is defined for all positive α).

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 α =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)=1(1−θt)α

for t < 1/θ.

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

M(t)=E(etX)=∫∞01Γ(α)θαe−x/θxα−1etxdx

Collecting like terms, we get:

M(t)=E(etX)=∫∞01Γ(α)θαe−x(1θ−t)xα−1dx

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

y=x(1θ−t)

Rearranging, we get:

x=θ1−θty and therefore dx=θ1−θtdy

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

Theorem. The mean of a gamma random variable is:

μ=E(X)=αθ

Proof. The proof is left for you as an exercise.

Theorem. The variance of a gamma random variable is:

σ2=Var(X)=αθ2

Proof. This proof is also left for you as an exercise.