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.)

Expectation and probability

Note that E[X] = E[Y], but these two random variables behave fairly differently.

1. What is the probability that Y is greater than 0? (I.e., what is P{Y > 0}?)

2. Let n = 500. Write a program that estimates the probability that X is greater than 0. Plot the probability distribution of X.

No one gets her/his own hat back

Let's try to approximate the probability that no one gets her/his own hat back in the random hat experiment.

1. In the random hat experiment what is the probability that no one get her/his own hat back? Give the exact answer.

This is the Sterling's approximation of factorial (See wikipedia entry)

$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$

2. Use Sterling's approximation to estimate the probability from question 1.

Binomial random variable

Consider the following standard experiment. We perform some experiment that has success probability p for n times. Let random variable X be the number of successful outcomes.

1. What is P{X = i}?

2. Define appropriate indicator random variables and use them to derive E[X].

3. (Bonus) Try to derive the same formula for E[X] directly from the definition of the expectation. (You may find Binomial theorem useful in this problem.)