ดูต้นฉบับสำหรับ 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<sub>i</sub>''' to be 1 if person '''i''' gets her/his hat back, and 0, otherwise.

1. What is '''E[X<sub>i</sub>]'''?

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 [http://en.wikipedia.org/wiki/Lazy_Susan 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 [http://en.wikipedia.org/wiki/Stirling%27s_approximation wikipedia entry])

:<math>n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n</math>

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 [http://en.wikipedia.org/wiki/Binomial_theorem Binomial theorem] useful in this problem.)