Birthday problem modified: Expected number of birthdays

Question

Given there are N people in one room and let us assume that they are born randomly on one of x(=365) days in the year. What is the expected number of different birthdays for N people?

Edit: I think the solution should be x-x(1-1/x)^n since (1-1/x)^n is the prob. that no-one has their birthday on a specific day and since all days are the same we should be able to multiply this probability with the number of days to get the expected number of "no birthday" days.

Answer 1

Hint: Let $X_1, X_2,\dots, X_{365}$ be indicator random variables taking value $1$ if their respective day of the year is the birthday of at least one of the $N$ people and $0$ otherwise.

How can we rephrase the meaning of the random variable $X=X_1+X_2+X_3+\dots+X_{365}$ in more natural language?

$X$ is the random variable denoting the number of distinct birthdays seen amongst our $N$ people.

If we were to look at $E[X]$, noting its relation to $X_1,X_2,\dots,X_{365}$, how can this help us for our problem?

$E[X] = E[X_1+X_2+\dots+X_{365}]=E[X_1]+E[X_2]+\dots+E[X_{365}]$ by linearity of expectation.

What more is left to complete the problem?

Calculate $E[X_i]$, notice the symmetry of the scenario, and complete the necessary arithmetic.