Let X∼Poisson(λ)X \sim \text{Poisson}(\lambda)X∼Poisson(λ). Find the value of the third factorial moment E[X(X−1)(X−2)]E[X(X-1)(X-2)]E[X(X−1)(X−2)].
λ3\lambda^3λ3
λ3+3λ2+λ\lambda^3 + 3\lambda^2 + \lambdaλ3+3λ2+λ
λ(λ−1)(λ−2)\lambda(\lambda-1)(\lambda-2)λ(λ−1)(λ−2)
3λ3\lambda3λ