Question

Let X1, X2, and X3 be mutually independent random variables with Poisson distributions having means 2, 1, 4, respectively. Find the moment generating function of the sum Y = X1+X2+X3

Answer 1

i am almost positive (almost) that the sum of the random variables is also poisson, with \(\lambda\) also the sum

Answer 2

meaning \(Y\) is poisson with \(\lambda =6\)

Answer 3

you know what the moment generating function for a possion random variable with mean \(\lambda\) is?

Answer 4

lol guess i can't add huh? make that \(\lambda=7\)

Answer 5

Since each \(X_i\sim\mathcal{P}(\lambda_i)\), you have
\[\large M_{X_i}(t)=\mathbb{E}(e^{X_it})=e^{\lambda_i(e^t-1)}\]and because
\[\large M_{X_1+X_2+X_3}(t)=M_{X_1}(t)M_{X_2}(t)M_{X_3}(t)\]it should be easy to determine the mgf for \(Y\).