182 Chapter 5: Special Random Variables
*5.7The Gamma Distribution
A random variable is said to have a gamma distribution with parameters (α,λ),λ>0,
α>0, if its density function is given by
f(x)=
λe−λx(λx)α−^1
(α) x≥^0
0 x< 0where
(α)=∫∞0λe−λx(λx)α−^1 dx=∫∞0e−yyα−^1 dy (by lettingy=λx)The integration by parts formula
∫
udv=uv−∫
vduyields, withu=yα−^1 ,dv=e−ydy,
v=−e−y, that forα>1,
∫∞0e−yyα−^1 dy=−e−yyα−^1∣∣
∣∣y=∞
y= 0+∫∞0e−y(α−1)yα−^2 dy=(α−1)∫∞0e−yyα−^2 dyor
(α)=(α−1) (α−1) (5.7.1)Whenαis an integer — say,α=n— we can iterate the foregoing to obtain that
(n)=(n−1) (n−1)
=(n−1)(n−2) (n−2) by letting α=n−1 in Eq. 5.7.1
=(n−1)(n−2)(n−3) (n−3) by lettingα=n−2 in Eq. 5.7.1
..
.
=(n−1)! (1)Because
(1)=∫∞0e−ydy= 1* Optional section.