Expectation Value of a function of Random variables in poisson distribution.

Question

Please Help me with this problem.

Let $X_i ∼ $Poi$(λ_i)$ be a sequence of independent random variables for $i = 1, 2, · · · , n.$

Define $~Y = X_1 + X_2 + \cdots + X_n~$. Then $E[Y^2]$ is ??

Answer 1

Hint:

Try to prove the following theorem:

If $X$ and $Y$ are independent with Poisson-distribution then also $X+Y$ has Poisson-distribution.

Answer 2

Thanks, @drhab, @Kezer for suggestions.

So, $E[Y^2] = Var[Y] + (E[Y])^2$ and If $ X_1 \ and\ X_2$ are independent with Poisson-distribution, then $X_1+X_2$ also has Poisson-distribution with $\lambda_{eff} = \lambda_{X_1} + \lambda_{X_2} $.

Hence as $ Y=X_1+X_2+\cdots +X_n$, Y has $\lambda_{eff}=\sum_{i=1}^n \lambda_i$. And as "Y" is itself a Poisson distribution, Variance and Expected value of Y are same as $\lambda_{eff}(Y)$.

Hence, the answer is, $E[Y^2]=\sum_{i=1}^n \lambda_i + ({\sum_{i=1}^n \lambda_i})^2 $.