"Emails arrive in an inbox according to a Poisson process of rate $\lambda$ emails per hour.
- Find the name and parameters of the conditional distribution of the number of emails that arrive in the first 2 hours of an 8-hour time period, given that exactly n emails arrive in that period.
- Each email is legitimate with probability $p$ and spam with probability $1-p$, independently. Find the name and parameters of the conditional distribution of the number of legitimate emails that arrive in an 8-hour time period, given that exactly s spams arrive in that period of time.
- Reading an email takes a random mount of time, with mean $\mu$ hours and a standard deviation $\sigma$ hours. These readings are i.i.d and independent of the email arrival process. Find the mean and variance of the total time it takes to read all the emails that arrive in an 8-hour time period." Introduction to Probability, second edition - Joseph Blitzstein, Jessica Hwang.
My solution is the following:
The number of emails in an 8-hour time interval is a Poisson process with rate of arrival $\lambda$. Given the number of arrivals, the arrival times of each email is Uniform distributed in the interval $(0-8)$. This means that the probability of an email to arrive in the interval (0-2) is $\frac{1}{4}$. Out of the total emails the number of the emails that arrive in the $(0-2)$ interval is $\sim Bin(n, \frac{1}{4})$.
Given that $s$ emails are spam, the number of emails that arrive in the 8-hour interval is equal to $s+k$ where k is the number of legitimate emails. In this case $P(Legit=k|Spam=s) = \binom{s+k}{k}p^kq^s$.
Total expectation can be found using Adam's Law: $E(E(T|N)) = 8\lambda\mu$ while variance can be found using the same principle $Var(T) = E(Var(T|N)) + Var(E(T|N)) = 8\lambda(\sigma + \mu^2)$
I don't have solutions for this problem and I'd appreciate if somebody could check my reasoning. I'm studying probability and statistics on my own and I don't have anybody to cross check my solution.