## Introduction to Probability Models

Lecture 20

Qi Wang, Department of Statistics

Oct 11, 2017

### Geometric Distribution Revisit

- $X \sim Geom(p)$
**The definition of $X$ **: the number of trials to get the first success
**Support:** $\{1, 2, \cdots\}$, NOTE: **NO ZERO!**
**Parameter:** $p$, the probability of success in one trial
**PMF:** $P_X(x) = p(1-p)^{x - 1}$
**Expected Value:** $E[X] = \frac{1}{p}$
**Variance:** $Var(X) = \frac{1 - p}{p^2}$
- Two import properties:
- Tail Probability formula: $P(X > k) = (1 - p)^k$
- Memoryless Property: $P(X > s + t| X > s) = P(X > t)$ and $P(X < s + t| X > s) = P(X < t)$

## Negative Binomial Random Variable

### Negative Binomial Random Variable

**The definition of $X$ **: the number of trials to get the $r_{th}$ success
**Support:** $\{r, r + 1, r + 2, \cdots\}$
**Parameter:**
- $p$: the probability of success in one trial
- $r$: success of interest

**PMF:** $P_X(x) = C_{r-1}^{x-1}p^r(1-p)^{x - r}$
**Expected Value:** $E[X] = \frac{r}{p}$
**Variance:** $Var(X) = \frac{r(1 - p)}{p^2}$
- $X \sim NegBin(r, p)$ or $X \sim NB(r, p)$

## Example 1

Pat is required to sell candy bars to raise money for the $6_{th}$ grade field trip.
He will ask his neighbors to buy a candy bar. There is a 40% chance of him selling
a candy bar to any neighbor that he asks. He has to sell 5 candy bars in all.
(Note: anyone purchasing will only buy ONE candy bar and the neighbors are independent of each other).

- What is the probability that he must ask 10 neighbors to sell all his candy bars?
- What is the probability that he asks fewer than 9 neighbors?
- How many neighbors does he expect to ask in order to sell all his candy bars?

### Example 2

The Plattsville Pluggers are a minor league baseball team.
Suppose that their ability to win any one game is 42% and games are independent of one another.

- What is the probability that it takes 14 games to get their $4_{th}$ win?
- What is the expected value and standard deviation of the number of games to get their $4_{th}$ win?
Their $25_{th}$ win? Their $1_{st}$ win?
- Knowing that the Pluggers got their $49_{th}$ win with 5 games remaining in the season,
what is the probability that they do NOT get 50 or more wins?

### Relationship Between Geometric Distribution and Negative Binomial Distribution

- Geometric a special case of the Negative Binomial when $r=1$
- $X_i \sim Geom(p), i = 1, 2, \cdots, r$, then $X = X_1 + X_2 + \cdots + X_r \sim NegBin(r, p)$
- $E[X] = E[X_1 + X_2 + \cdots + X_r] = E[X_1] + E[X_2] + \cdots + E[X_r] = \frac{r}{p}$
- $Var(X) = Var(X_1 + X_2 + \cdots + X_r) = Var(X_1) + \cdots + Var(X_r) = \frac{r(1 - p)}{p^2}$