probability and statistics

Question

Suppose that n people throw their hats in a box and then each picks up one hat at

random. Let X be a random variable which is defined as the number of people that get

back their own hat.

a.) Find probability mass function px(X) of X for n=3, and calculate expected

value of X by using px(X)

b.)Let for Xi=1 if the ith person selects his/her own hat, and for Xi=0 otherwise

show that P(Xi=1)=1/n i=1,2,....n

Answer 1

There are 6 possible arrangements: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1),(3, 1, 2), (3, 2, 1)

We have 3 matches, 1 match, 1 match, 0 matches, 0 matches, 1 match

Thus, we have 0 matches 2 times, 1 match 3 times, and 3 matches 1 time

P(0) = 2/6 = 1/3 P(1) = 3/6 = 1/2 P(2) = 0/6 = 0 P(3) = 1/6

E(x) = the sum of x P(x) = 0 * 1/3 + 1 * 1/2 + 2 * 0 + 3 * 1/6 = 0 + 1/2 + 0 + 1/2 = 1

b) There are n! possible arrangements. If the i'th person selects the i'th hat, the other n-1 people, can have any possible ordering of the other n-1 hats, so there are n-1! possible combinations. Thus, P(X_i = 1) = n-1!/n! = n-1!/(n*n-1!) = 1/n