6.11 The gamma function : [0,) [0,) is defined as () = 0 t 1et dt, and is one of the

6.11 The gamma function Γ : [0,∞) → [0,∞) is defined as

Γ(α) =  ∞

0 t

α−1e−t dt, and is one of the most useful mathematical functions. In statistics it is the basis of the gamma and beta distributions (see the following problems). The gamma function has several interesting properties which you will prove next.

1. Show that Γ(α + 1) = αΓ(α),α> 0.

2. Show that Γ(n)=(n − 1)! for any integer n > 0.

3. Show that Γ( 1 2 ) = √π and accordingly calculate the general formula for Γ( n 2 )

for any integer n > 0.

4. Show that f(t) = t

α−1e−t

Γ(α) 1{t∈(0,∞)}, is a proper density function for all α > 0.

6.12 The Gamma(α, β) distribution may be obtained from the gamma function in exercise 6.11 but it has two parameters α also called the shape parameter and β

also called the scale parameter due to their influence on the peakedness, respectively, spread of the distribution. Specifically, its density is f(x) = 1

βαΓ(α)

xα−1e−x/β.