I'm really struggling with understanding how to use the Law of Total Probability in proof questions such as this one. Any help would be hugely appreciated.
The number of emails received in a day has a Poisson distribution with mean 200. On that day, the probability of one email being spam is denoted by p. Using the Law of Total Probability, show that the number of spam emails received that day also follows a Poisson distribution. Is it possible to identify the value of p from this?
Best Answer
The information that you are given is about the behavior of the number of spam emails, GIVEN the total number of emails. Although it isn't explicitly spelled out, if $N$ is the (stochastic) number of emails received, then the number of spam emails follows the Binomial distribution with $N$ trials and probability of "success" (spam) $p$.
How to use this? The law of total probability says that if $K$ is the number of spam messages, then $$ P(K=k)=\sum_{n=0}^{\infty}P(K=k\mid N=n)\cdot P(N=n). $$ We already know that $$ P(N=n)=e^{-200}\frac{200^n}{n!}. $$ And, condition on $N=n$, we know the distribution of $K$: in particular, it is $\text{Bin}(n,p)$. You can use this to compute, for any $n$, $P(K=k\mid N=n)$. From there, it is just a matter of completing the summation.