Question

(Hat Check problem). n people enter the restaurant and put their hats at the reception. Each person gets a random hat back when going back after having dinner. Find the expected value and variance of the number of people who get their right hat back.

Answer 1

E[X] = 1

Var[X] = 1

Explanation:

We are told that n people enter the restaurant and put their hats at the reception.

This means that there are n hats.

Also, we are told that eeach person gets a random hat back when going back after having dinner.

Thus, each person will pick a hat uniformly at random and will get a right hat back with the probability: 1/n.

The expectation is linear even though the random variables are dependent. Thus, the expected mean of the total number of persons who get their right hat back is: E[X] = n × 1/n = 1

Now, the variance in hat problem is given by;

Var[X] = E[X²] - E[X]²

This gives;

Var[X] = E[X²] - 1

Now, E[X²] is expressed as;

E[X²] = n(1/n) + n(n - 1)•(1/n)•(1/(n - 1))

E[X²] = 1 + 1

E[X²] = 2

Thus;

Var[X] = 2 - 1

Var[X] = 1