Linear Combinations (I think)

[Imum Coeli]

Hi I have a question and I'm afraid that i missed something.

Q: A retail grocery merchant figures that her daily gain from sales $$\displaystyle X$ is a normally distributed random variable with $\displaystyle \mu =50$ and $\displaystyle \sigma = 3$. $\displaystyle X$ can be negative if she has to dispose of enough perishable goods. Also, she figures daily overhead costs are $$\displaystyle Y$, which have an Erlang distribution of order $\displaystyle n=4$ and common rate $\displaystyle \lambda = 0.5$. if $\displaystyle X$ and $\displaystyle Y$ are independent, find the expected value and variance of her daily net gain.

A:
Let $\displaystyle P$ be her daily net gain then

$\displaystyle P = X - Y$

So the expected valued of $\displaystyle P$ is

$\displaystyle E[P] = E[X]-E[Y]$

and since $\displaystyle X \sim \text{N}(50,3) \implies E[X] = 50$ and $\displaystyle Y \sim \text{Erl}(4,0.5) \implies E[Y]= \frac{4}{0.5} = 8$ then

$\displaystyle E[P]= 42$ (coincidentally her expected profit is the meaning of life).

Now the variance of $\displaystyle P$ is

$\displaystyle \text{var}(P) = var(X) + var(Y)$ since independent.

and $\displaystyle X \sim \text{N}(50,3) \implies \text{var}(X)= 3^2=9$ and $\displaystyle Y \sim \text{Erl}(4,0.5) \implies \text{var}(Y)= \frac{4}{(0.5)^2} = 16$ then

$\displaystyle \text{var}(P) = 25$.


(Sorry if this seems like a stupid question but we have just had a baby two days ago and I need to talk about this problem. Hopefully I don't look too much like an idiot )