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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 $X$ ?

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  • $\begingroup$ 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? $\endgroup$ – Nap D. Lover Nov 24 '18 at 0:16
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The method you used is correct. Notice that adding a constant does not change the variance of a random variable. Therefore $$ Var(X) = 2000^2 * Var(C) $$

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  • $\begingroup$ Hey! I checked the answer given at the back of the text and it is 16000. That is why I say that it is not correct. I don’t know how they’re getting 16000 though. $\endgroup$ – user601297 Nov 24 '18 at 9:47

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