## Introduction to Probability Models

Lecture 18

Qi Wang, Department of Statistics

Oct 2, 2017

## Reminders

- The fourth quiz will be on this
**Wednesday, Oct 4**
- The third homework is posted and due on
**Oct 16**
- No class on
**Oct 6** as Exam 1 compensation

### Example 1

Consider a game where we will roll a fair die. We will roll it until we get a $5$.
What is the probability that it will take $3$ rolls to get the $5$?

Think about:

- Are the subsequent rolls independent?
- What about the probability of success on each roll?
- Do we have a set number of trials?

## GEOMETRIC RANDOM VARIABLE

### Characteristics of the Geometric Distribution

**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:** $Var(X) = \frac{1 - p}{p^2}$
- $X \sim Geom(p)$

## Example 2

Suppose Dunphy is really bad at tossing a Frisbee and unfortunate hits people walking by at a rate of 1 out of every 5 people.

- What is the probability that his first accidental hitting is the $6_{th}$ or $7_{th}$ person to walk by?
- What is the probability that more than 7 people walk past before he hits one with the Frisbee?

### Important Properties for the Geometric Distribution

- Tail Probability formula: $P(X > k) = (1 - p)^k$
- Memoryless Property: $P(X > s + t| X > s) = P(X > t)$ and $P(X < s + t| X > s) = P(X < t)$

### Example 2 continued

- 3. Four people have walked past Dunphy and none have been hit by a Frisbee. What is the probability that at most 9 walk by until the first person is hit by a Frisbee?
- 4. Four people have walked past Dunphy and none have been hit by a Frisbee. What is the probability that at least 10 walk by until the first person is hit by a Frisbee?

### Example 3

Shaq is shooting free throws in the gym. He intends to stay until he makes one. His probability of making one on any free throw is 0.527. Let X be the number of attempts until he makes one.

- Distribution, parameter, support?
- Expected number of shots until he makes one?
- Probability he makes his first shot on the $4_{th}$ try?
- Probability it takes him at least 4 shots to make $1_{st}$?
- Probability it takes him exactly 4 shots if he already missed the first?
- Probability it takes him at least 4 shots if he already missed the first?