I've been reading up on Poisson Distributions and have come across the following problem. My doubts are in Bold:
What's the probability that in a room of n people, nobody shares the same birthday?
Q.1. Do we consider that N(the number of successes) follows a Poisson Distribution as there is a fixed time interval of 365 days?
The answer states that $X_{ij}= 1 $(if person i and j share the same birthday and is 0 otherwise)
And then the answer states
$$ E[N] = E[\sum\limits_{i<j} X_{ij}] = \binom{n}{2}\frac{1}{365}$$
Q.2 I don't understand how we know that $ E[N] = E[\sum\limits_{i<j} X_{ij}] $ and that $E[\sum\limits_{i<j} X_{ij}] = \binom{n}{2}\frac{1}{365}$
Thank You
I think I can guess how the answer you're looking at works. Firstly, note that the probability that person i and person j share the same birthday is 1/365.
$E[N] = E[\sum_{i<j} X_{ij}]$ follows just the definition of $N$. $N$ is the number of people that share a common birthday. Therefore, to calculate $N$, ask all pairs whether they share a birthday. Whenever a pair does $X_{ij}=1$, when it doesn't $X_{ij}=0$. Adding up all $X_{ij}$ where the pairing is unique gives $N$. chris and mark share a birthday => mark and chris share a birthday, but we only want to count that once as a pair. The condition $i<j$ ensures we don't count pairs twice. That gives us this first result. Next result:
$E[\sum_{i>j}X_{ij)}]$ is an expectation of a sum of random variables, so we can distribute the expectation over all terms in the sum. $E[\sum_{i<j}X_{ij}]=\sum_{i<j}E[X_{ij}]$. The definition of X_(ij) gives a simple expectation by probability mass: $E[X_{ij}]=P(X_{ij}=0)*0+P(X_{ij}=1)*1=P(X_{ij}=1)$. but that's just the probability that two people share a birthday which we know is 1/365.
Therefore: $E[N]=\sum_{i<j}\frac{1}{365}=\frac{Q}{365}$ where $Q$ is the number of terms in the sum. How many terms are there in the sum? That's just how many pairs of people you can form which is $C(n,2)$. that gives the second result.
As regards use of the poisson distribution, a little bit of poking around the internet suggests that N is indeed well approximated by the poisson distribution, but this seems to be something which is just understood to be true. The parallels between this and people arriving into a queue randomly over some time interval with some mean rate are not very clear to me. In particular note that the mean rate of birthday sharing we've calculated depends non-linearly on the interval we're considering. Anyway, we've calculated the mean therefore we've fully parameterised our approximate model and we can approximate $P(N=0)$ - nobody shares a birthday - using $N \sim Po(\frac{1}{365}C(n,2))$. More importantly we have a general expression for $P(N=n)$