Birthday problem: expected number of people with a birthday on a given date?

Question

Here's another formulation of a birthday problem: given $n$ people, and $m$ days, how to calculate the expected number of people having a birthday on any single day? I.e., given a random variable $X$ denoting a number of people with a birthday on a given date, how to find $\mathbb{E}[X]=\sum_{k=0}^{n}k\mathbb{P}[X=k]$?

Answer 1

Let $X_i = 1$ if person $i$ has a birthday on fixed day $b$, and $0$ otherwise. We want to know the value of $X = X_1 + X_2 + X_3 + ... + X_n$ on average.

Then by linearity of expectation:

$$E[X] = E\left[\sum_{i=1}^{n}X_i\right] = \sum_{i=1}^{n} E[X_i] = \sum_{i=1}^{n} \dfrac{1}{m} = \dfrac{n}{m}$$