What is the expected number of days of the year that are a birthday of at least one of n people?

Question

I am new to probability and am just learning about the birthday problem. There is a sample problem that asks among a group of $n$ unrelated people, what is the expected number of days of the year that are a birthday of at least one person. I am not sure how to approach this. Any help would be appreciated. Thanks.

Answer 1

Introduce for typical year of $365$ days long Bernoulli random variables $X_1,\ldots, X_{365}$. Let $X_i=1$ if day $i$ is a birthday of at least one person and $X_i=0$ else. For each person it is equally likely to have a birthday at any day of a year, so nobody born at day $i$ with probability $$ \mathbb P(X_i=0) = \left(\frac{364}{365}\right)^n. $$ Then $X_i=1$ with probability $1-\left(\frac{364}{365}\right)^n$.

The number of days with at least one birthday is $$ X=X_1+\ldots+X_{365}. $$ Find expectation of $X_1$ and use linearity of expectation to find $\mathbb E[X]$.