Stat 5101 Lecture Notes - School of Statistics

4.3. THE BETA DISTRIBUTION 117course we will always use (4.8), and following Lindgren we will use the notationGam(α, λ) to denote the distribution with density (4.8). We will call λ theinverse scale parameter or, for reasons to be explained later (Section 4.4.3), therate parameter. The fact that (4.8) must integrate to one tells us∫ ∞x α−1 e −λx dx = Γ(α)0λ α .We can find the mean and variance of the gamma using the trick of recognizinga probability density (Section 2.5.7).E(X) =∫ ∞0= λαΓ(α)= λαΓ(α)xf(x | α, λ) dx∫ ∞0Γ(α +1)λ α+1x α e −λx dx= α λ(we used the recursion (4.5) to simplify the ratio of gamma functions). SimilarlyE(X 2 )=∫ ∞0x 2 f(x|α, λ) dx∫ ∞= λαx α+1 e −λx dxΓ(α) 0= λα Γ(α +2)Γ(α) λ α+2(α +1)α=λ 2(we used the recursion (4.5) twice). Hencevar(X) =E(X 2 )−E(X) 2 = (α+1)α α) 2 α−(=λ 2 λ λ 2The sum of independent gamma random variables with the same scale parameteris also gamma. If X 1 , ..., X k are independent with X i ∼ Gam(α i ,λ),thenX 1 + ···+X k ∼Gam(α 1 + ···+α k ,λ).This will be proved in the following section (Theorem 4.2).4.3 The Beta DistributionFor any real numbers s>0 and t>0, the functionh(x) =x s−1 (1 − x) t−1 ,0