(Solved): 6. Let \( X \sim \) Poisson \( (\lambda) \). Find \( E[X(X-1)(X-2)] \) using the following steps: ...

6. Let \( X \sim \) Poisson \( (\lambda) \). Find \( E[X(X-1)(X-2)] \) using the following steps: - Write using definition \( E[X(X-1)(X-2)]=\sum_{k} k(k-1)(k-2) P(X=k) \) - Change the summation to start from \( k=3 \), because \( k(k-1)(k-2)=0 \) when \( k=0,1,2 \) - Rewrite the summation using a new index \( i=k-3 \) - Simplify the summation using the identity \[ \sum_{i=0}^{\infty} \frac{x^{i}}{i !}=e^{x} \] Hence, find \( E\left(X^{3}\right) \).