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