Let X1,X2,X3X_1, X_2, X_3X1,X2,X3 be i.i.d. Poisson(λ)\text{Poisson}(\lambda)Poisson(λ) random variables. Find the conditional expectation E[X12∣X1+X2+X3=n]E[X_1^2 \mid X_1 + X_2 + X_3 = n]E[X12∣X1+X2+X3=n].
n2+2n9\frac{n^2 + 2n}{9}9n2+2n
n2+n9\frac{n^2 + n}{9}9n2+n
n29\frac{n^2}{9}9n2
n2+3n9\frac{n^2 + 3n}{9}9n2+3n