# Discrete Random Variables

## Hypergeometric Distribution (Optional)

There are five characteristics of a hypergeometric experiment:
1. You take samples from two groups.
2. You are concerned with a group of interest, called the first group.
3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of 10 players.
4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a woman first is 
 . The probability of picking a man second is 
  if a woman was picked first. It is 
  if a man was picked first. The probability of the second pick depends on what happened in the first pick.
5. You are not dealing with Bernoulli trials.
The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest.

### Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function

X ~ H(r, b, n)
Read this as X is a random variable with a hypergeometric distribution. The parameters are r, b, and n: r = the size of the group of interest (first group), b = the size of the second group, n = the size of the chosen sample.

### Chapter Review

A hypergeometric experiment is a statistical experiment with the following properties:
1. You take samples from two groups
2. You are concerned with a group of interest, called the first group
3. You sample without replacement from the combined groups
4. Each pick is not independent, since sampling is without replacement
5. You are not dealing with Bernoulli trials
The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. The distribution of X is denoted X ~ H(r, b, n), where r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample. It follows that n ≤ r + b. The mean of X is μ = 
  and the standard deviation is σ = 
 .

### Formula Review

X ~ H(r, b, n) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample.
X = the number of items from the group of interest that are in the chosen sample, and X may take on the values x = 0, 1, . . . , up to the size of the group of interest. The minimum value for X may be larger than zero in some instances.
n ≤ r + b
The mean of X is given by the formula μ = 
  and the standard deviation is = 
 .
Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.

### HOMEWORK
