Probstat/week4 practice 1

Two experiments

In this section, we shall analyze two experiments.

Random hats

A group of n people, each wearing a different hat, go to the museum. They have to leave their hats at the entrance. When they get back, each gets a random hat back. We are interested in the number of people who get their own hat back.

Let random variable X be the number of people who get their own hat back. As a typical way of using linearity of expectation, we shall define an indicator random variable X_i to be 1 if person i gets her/his hat back, and 0, otherwise.

1. What is E[X_i]?

2. What is E[X]? (Show your work.)

Dinner on a circle table

The same group of n people go into a Chinese restaurant. They sit on a circular table with a circular turntable (see wikipedia article). Each person orders one different dish and gets her/his order exactly in front of her/him. To make a fun dinner, they decide to randomly rotate the turntable so that each one of them will hopefully get a random dish.

Let random variable Y be the number of people who get their own dish after the random rotation.

1. What is E[Y]? (In this case, you probably don't need to use the linearity of expectation.)