Variance of $2X_1 +X_2+3X_3$ with $X_i \sim \operatorname{Poisson}(i x \lambda)$

Question

$X_1, X_2, X_3$ are independent random variables such that $X_i \sim \operatorname{Poisson}(i x \lambda)$, $i=1,2,3$. What is the variance of $2X_1 + X_2 +3X_3$?

I know how to find expectation/variance etc, but would really appreciate some help as to how to go about answering this question.

Answer 1

Hint:

• use the variance of a Poisson distribution three times
• use the variance of a scaled distribution two times
• use the variance of a sum of independent random variables

Answer 2

When $X_1, X_2, \ldots, X_n$ are independent random variables, we have $$\operatorname{Var}\left[\sum_{i=1}^n c_i X_i\right] = \sum_{i=1}^n c_i^2 \operatorname{Var}[X_i]$$ for any fixed scalar constants $c_1, c_2, \ldots, c_n$. So for example, with $n = 2$, the above formula reduces to $$\operatorname{Var}[c_1 X_1 + c_2 X_2] = c_1^2 \operatorname{Var}[X_1] + c_2^2 \operatorname{Var}[X_2].$$ The only required condition is that the random variables are independent: they need not follow the same parametric distribution.

For the expected value, it should be noted that the independence condition is not required, because expectation is a linear operator. Thus $$\operatorname{E}\left[\sum_{i=1}^n c_i X_i\right] = \sum_{i=1}^n c_i \operatorname{E}[X_i]$$ whether or not the random variables are independent or identically distributed.