The number of emails I receive at work per hour has Poisson distribution: $X \text{~} Po(4)$.
Tomorrow I will work for T hours, where T is uniformly distributed on {$2,3,4,5$} - with number of hours worked independent of number of emails received.
Find the variance of the total number of emails received tomorrow.
Would I find a way to calculate the joint distribution, then find the variance of that distribution? Or is there a formula for applying the uniform distribution on poisson that I am unaware of?
By the law of total expectation: $\mathbb{E}(Y)=\mathbb{E}(Y|2)\mathbb{P}(T=2) + \mathbb{E}(Y|3)\mathbb{P}(T=3) + \mathbb{E}(Y|4)\mathbb{P}(T=4) + \mathbb{E}(Y|5)\mathbb{P}(T=5)$
$= \mathbb{E}(X \text{~} Po(8))*1/4 + \mathbb{E}(X \text{~} Po(12))*1/4 + \mathbb{E}(X \text{~} Po(16))*1/4 + \mathbb{E}(X \text{~} Po(20))*1/4$
$=\frac{8}{4} + \frac{12}{4} +\frac{16}{4} +\frac{20}{4} = 14 $
Similarly,
$\mathbb{E}(Y^2)=\mathbb{E}(Y^2|2)\mathbb{P}(T=2) + \mathbb{E}(Y^2|3)\mathbb{P}(T=3) + \mathbb{E}(Y^2|4)\mathbb{P}(T=4) + \mathbb{E}(Y^2|5)\mathbb{P}(T=5)$
Since $\mathbb{E}(X^2) = \lambda^2 + \lambda$ for Poisson distribution, we have:
$\mathbb{E}(Y^2)=230$
Therefore, $Var(Y) = 230 - 14^2 = 34$