Using R Studio write code for the following:

### Diploma problem (hat-check) You most likely heard about this problem before. Smith College is a residential women's liberal arts college in Northampton, MA that is steeped in tradition. One such tradition is to give each student at graduation a diploma at random. At the end of the ceremony, a diploma circle is formed, and students pass the diplomas that they receive to the person next to them, and step out once they've received their diploma. What is the expected number of students who receive their diplomas in the initial disbursement?

The analytic solution (of the expect value) is easy to derive. Let $X_i$ is the event that $ith$ student receives their diploma then $E(X_i)=1/n$, for all i (the diplomas are uniformly distributed). n is the number of diplomas (students). Thus, if $Y$ is the sum of all the events $X_i$, then $E(Y)=1$. It is sometimes kind of surprising that the expected number of students receiving their diplomas in the initial disbursement does not depend on $n$. The variance can be more difficult to derive since $X_i$ are dependent.

We will solve the problem using simulations with R. Simulate the problem and find the expected value and the variance of the number of students who receive their diplomas in the initial giving.