Assume I have products available to buy and I get 4 ($\lambda = 4$) customers (poisson distributed) wanting to buy some of them everyday for $10 each.
The poisson distribution is
$$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$
I know that the expectation of a poisson distribution is the $\lambda$, so then is my expected profit going to be $40 everyday or do I have to do something else in order to calculate the expectation of profit?
If you want more detailed solution:
$E[\text{profit}] = \sum^{\infty}_{k = 0} 10 \cdot k \cdot \frac{\lambda ^ k e^{-\lambda}}{k!} = 10 \cdot \sum^{\infty}_{k = 0}\frac{\lambda ^ k e^{-\lambda}}{k!} = 10 \cdot \lambda = 40.$
As to how $ \sum^{\infty}_{k = 0}\frac{\lambda ^ k e^{-\lambda}}{k!} = \lambda$, you can find the proof here.
Hope it helped!