## Introduction to Probability Models

Lecture 16

Qi Wang, Department of Statistics

Sep 27, 2017

## Hypergeometric Distribution

### Hypergeometric Distribution

**The definition of $X$**: the number of success in $n$ trail without replacement from a finite population of size N that contains exactly M objects with that feature.
**Support:** $\{0, 1, 2, \cdots, n\}$ or $\{0, 1, 2, \cdots, M\}$
**Parameters:**
- $N:$ Population size
- $M:$ Number of possible successes
- $n:$ Number of trials

**PMF:** $P_X(x)\frac{C_x^M C_{n-x}^{N - M}}{C_n^N}$
**Expected Value:** $E[X] = n \frac{M}{N}$
**Variance:** $n \frac{M}{N} (1 - \frac{M}{N}) \frac{N-n}{N-1}$
- $X \sim Hyper(N, n, M)$

### Example 1

There are 100 identical looking 52" TVs at Best Buy in Costa Mesa, California. Let 10 of them be defective. Suppose you want to buy 8 of the aforementioned TVs (at random).

- What is the probability that you don't get any defective TVs? Identify the distribution parameters and support.
- Given that we purchase at least one defective TV, what is the probability that you purchase fewer than 3 defective TVs?
- What is the expected number of defective TVs that you will purchase?
- Find the standard deviation of the number of defective TVs that you purchase.

### Example 2

An experiment consists of shuffling a standard deck of 52 cards and then dealing a 5 card hand. Let Y denote the number of diamonds in the hand.

- Identify the distribution of Y and give its parameter(s) and support. Find the probability that Y is 2.
- Suppose instead of using 1 deck, we mix together 1,000 decks. The cards are shuffled and 5 are dealt into a hand. Let D denote the number of diamonds in the hand. Find the exact probability that you get 2 diamonds.

###
The Binomial Approximation to the Hypergeometric

In probability, we can use some distributions to approximate others.

- If $X\sim Hyper(N, n, M)$ AND $N > 20n$, then $X \sim Bin(n, p = \frac{m}{N})$
- With a large enough population, sampling without replacement will also get a Binomial.
- So back to Example 2, is an approximate distribution appropriate for D, why or why not?
- Use that approximation to find P(D = 2). What is the distribution, parameter(s) and support for this approximating distribution?