MomentsE[aX+b]=aE[X]+b\mathbb{E}[aX+b] = a\mathbb{E}[X] + bE[aX+b]=aE[X]+b V[aX+b]=a2VX\mathbb{V}[aX+b] = a^2 \mathbb{V}{X}V[aX+b]=a2VX X∼γ(α,θ)X \sim \gamma(\alpha, \theta)X∼γ(α,θ), E[Xk]=Γ(α+k)Γ(α)θk\mathbb{E}[X^k] = {\Gamma(\alpha+k) \over \Gamma(\alpha)} \theta^kE[Xk]=Γ(α)Γ(α+k)θk Γ(α+1)=αΓ(α)\Gamma(\alpha+1) = \alpha \Gamma(\alpha)Γ(α+1)=αΓ(α), Γ(1)=1, Γ(12)=π\Gamma(1) = 1,~\Gamma({1 \over 2}) = \sqrt{\pi}Γ(1)=1, Γ(21)=π
