Expectation of sum of independent Poisson distributions

Question

I have three independent Poisson distributions:

• $X_1 \sim \mathcal{P}(15)$,

• $X_2 \sim \mathcal{P}(21)$ &

• $X_3 \sim \mathcal{P}(10)$.

I wish to find the Expectation and Variance of $X_1 + X_2 + X_3$.

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For the expectation, my first intuition was to add up the $\lambda$s i.e. $15+21+10$ since they are independent. Similar reasoning for the variance. However, I'm not sure if my reasoning is correct.

I would appreciate any help.

Answer 1

Your three random variables are Poisson distributed. The sum of these random variables will also be Poisson. So your intuition was correct about the expectation.

$X1 + X2 + X3 \sim \mathcal{P}(15 + 21 + 10)$

https://en.wikipedia.org/wiki/Poisson_distribution#Properties (see the fourth section under properties of the Poisson distribution)

Now that you know the distribution of your sum you can take it from here!

Answer 2

$E(\sum_{i=1}^n X_{i}) = \sum_{i=1}^n E(X_{i})$

$V(\sum_{i=1}^n X_{i}) = \sum_{i=1}^n V(X_{i})$ since $COV(X_i,X_j)=0$ due to independence.

So you are correct