## Introduction to Probability Models

Lecture 17

Qi Wang, Department of Statistics

Sep 28, 2017

### Poisson Distribution

**The definition of $X$**: the number of success per $\underline{\hspace{1cm}}$, and $\underline{\hspace{1cm}}$ can be time, length, space unit and so on
**Support:** $\{0, 1, 2, \cdots\}$
**Parameters:** $\lambda$, the average success rate per $\underline{\hspace{1cm}}$
**PMF:** $P_X(x) = \frac{e^{-\lambda} \lambda^x}{x!}$
**Expected Value:** $E[X] = \lambda$
**Variance:** $Var(X) = \lambda$
- $X \sim Poisson(\lambda)$

### Rule for using a POISSON to approximate a BINOMIAL random variable:

If $X \sim Bin(n, p)$ with **$n>100$ AND $p<0.01$**, we can approximate $X$ by $X^\star \sim Poisson(\lambda = np)$

## Example 1

Suppose earthquakes occur in the western US on average at a rate of 2 per week. Let $X$ be the number of earthquakes in the western US this week.

- Find the probability that $X$ is 3. What distribution and parameter(s) are you using?
- What is the probability that there are at least 2 earthquakes in a week in the western US?
- What is the expected number of earthquakes and the standard deviation of the number of earthquakes in the western US in a week?

Now consider a month. Let Y be the number of earthquakes in the western US this month(assume that 1 month is equivalent to 4 weeks).

- Find the probability that Y is 12. What distribution and parameter(s) are you using?
- Let $Z$ be the number of weeks in a 4 week period that have a week with 3 earthquakes in the western US. Find the probability that $Z$ is 4. Is this the same as the probability that $Y$ is 12? Does this make sense?

### Example 2

Customers arrive at the UPS store randomly and independently at a rate of 15 per hour.

- What is the probability that 45 customers arrive between 11:30 am and 3:00 pm? What distribution and parameter(s) are you using? What is the support?
- What is the probability that 45 customers arrive between 11:30 am and 3:00 pm AND 10 customers arrive between 3:00 pm and 3:45 pm?

### Example 3

Flaws on an old computer tape occur on average every 1200 feet. You have an old computer tape roll that is 4800 feet long.

- What is the probability that there is at least one flaw on that roll?
- You know that there is at least one flaw on the roll. Knowing this, what is the probability that there are 2 or 3 flaws on the roll?

### Example 4

A certain disease occurs in 7 out of 5000 people. We will conduct a study and take a sample of 1000 people. What is the probability that no one in the sample has the disease?