I was given a homework question that is stated in the title. Although I have a conflict with the solution provided, and was wondering if you could help me understand why the solution is correct or if it is indeed incorrect.
Define $X$ to be number of distinct birthdays.
The answer given is to set up a RV $X_i$ which is $1$ if the ith day is a birthday or $0$ otherwise, where:
$P(X_i = 1) = P(\text{at least one person has birthday on day i}) = 1- P(\text{no one has birthday on this day}) = 1 - \frac{364}{365}^{100}$. And so $\mathrm{E}X_i = 1 - \frac{364}{365}^{100}$
Thus $\mathrm{E}X =\mathrm{E}[X_1 + X_2 \dots X_{365}] = 365\left (1 - \frac{364}{365}^{100} \right)$
I think this is incorrect, however. The reason being is that it seems like they are calculating the expected number of birthdays not the expected number of distinct birthdays.
The answer that I think is correct is to define $X_i$ as $1$ if the ith day is a distinct birthday and $0$ otherwise. Then:
$P(X_i = 1) = 100 \times \left(\frac{1}{365}\right)\left(\frac{364}{365}\right)^{99}$.
Thus $\mathrm{E}X =\mathrm{E}[X_1 + X_2 \dots X_{365}] = 365 \times 100 \times \left(\frac{1}{365}\right)\left(\frac{364}{365}\right)^{99} = 100 \times \left(\frac{364}{365}\right)^{99}$.
This has been bothering me for quite some time. Any help would be great.