# Thread: Linear combination of poisson random variables

1. ## Linear combination of poisson random variables

Hi, can someone help me with this question:

Let X~Poisson(1), Y~Poisson(2), define T = X + Y.
a) Show that T~Poisson(3)
b) Find the joint probability function of X and T
c) Find the conditional distribution of X given T = n
d) Compute Cov(X,T) and the correlation coefficient.

I have no problems with a). But I got stuck at b) because I get that the jpf of X and T, f(X,T) = P ( X = x , T = t) = f(T) ...?! Which makes no sense. Any help would be appreciated!

2. Assuming independence....

Use moment generating functions to show (a).

$P(X=a,T=b)=P(X=a, Y=b-a)=P(X=a)P(Y=b-a)$

$= \left({e^{-1}1^a\over a!}\right) \left({e^{-2}2^{b-a}\over (b-a)!}\right)$

for (c) just use the definition of conditional probabilities.

(d) Cov(X,X+Y)=Cov(X,X)+Cov(X,Y)=V(X)+0=1.

3. ah yes, thank you very much! So the difference between a) and b) is that in b) we are not summing all the marginal probabilities of x and y, because we have a constraint on what X is, correct?