# Poisson Distribution Variance Problem

Here is the question-

The number of computer servers that break down during a month is a Poisson Random Variable with parameter $$\lambda = 2$$. The cost of repairing one server is 2000 and also there is a fixed overhead cost of 10000 given as salary to the technician. If $$X$$ is the total expenditure made on repairs during a month find expectation and variance of $$X$$.

I managed to calculate the expectation as follows

$$\mathbb{E}[X] = 10000 + 2000*\mathbb{E}[C]$$ where $$C$$ is the number of computer servers that break down.

$$\mathbb{E}[X] = 10000 + 2000*2 = 14000$$

However I can’t seem to calculate the variance of $$X$$, I know that $$Var(C) = 2$$.

The following method I know is wrong $$Var(X) = (2000)^2*Var(C) = (2000)^2*2$$ So, how to calculate the same for $$X$$ ?

• the variance of $c+N$ where $N$ is a RV and $c$ is a constant is equal to the variance of $N$. Extremely useful and basic property of variance. Can you combine this with the other property you listed for scaled RVs to finish? – Nap D. Lover Nov 24 '18 at 0:16

The method you used is correct. Notice that adding a constant does not change the variance of a random variable. Therefore $$Var(X) = 2000^2 * Var(C)$$