Question: 2016 Fall Final - 20
Let $n \geq 2$ be an integer and consider a group $P_1,P_2,\dots,P_n$ of $n$ people.
Each of these people has a uniformly random birthday, which is independent of the birthdays of the
other people. We ignore leap years; thus, the year has 365 days.
Define the random variable $X$ to be the number of unordered pairs $\{P_i,P_j\}$ of people that have the same birthday.
What is the expected value $\mathbb{E}(X)$ of $X$?
Hint: Use indicator random variables.
Define the random variable $X$ to be the number of unordered pairs $\{P_i,P_j\}$ of people that have the same birthday.
What is the expected value $\mathbb{E}(X)$ of $X$?
Hint: Use indicator random variables.
(a)
$\left( \frac{1}{365} \right)^2 \cdot n^2$
(b)
$\frac{1}{365} \cdot n^2$
(c)
$\frac{1}{365} \cdot {n \choose 2}$
(d)
$\left( \frac{1}{365} \right)^2 \cdot {n \choose 2}$