## Introduction to Probability Models

Lecture 26

Qi Wang, Department of Statistics

Oct 25, 2017

### Reminder

- Homework 4 is due on
**Oct 30**
- Exam 2 will be from
**8:00pm to 9:30pm** on **Thursday, November 2nd**

### Example 1

At a high school track and field tournament, Mark’s high jumps vary evenly from 1.8 meters to 2.15 meters,
while Dan’s high jumps vary evenly from 1.75 to 2.3 meters.

- Let M be the length of one of Mark’s high jumps. What are the distribution and parameter(s) of M?
- What is that the probability that Mark jumps between 1.88 and 2.05 meters?
- Which jumper’s jumps has the smaller standard deviation?
- What is the probability that one of Dan’s high jumps is exactly 2.0 meters?
- What length cuts off the highest 25% of Dan’s high jumps?

### Exponential Random Variable

**The definition of $X$ **: The waiting time until the first success
**Support:** $X \in [0, +\infty)$ or $X \ge 0$
**Parameter:** $\mu$, the average amount of time for one success
**PDF:** $f_X(x) = \frac{1}{\mu} e^{-\frac{x}{\mu}}$, for $ x \ge 0$
**CDF: **
\[
F_X(x)= P(X \le x) = \left\{
\begin{array}{ll}
0, x < 0\\
1 - e^{-\frac{x}{\mu}}, x \ge 0
\end{array}
\right.
\]
**Expected Value:** $E[X] = \mu$
**Variance:** $Var(X) = \mu^2$
**Notation:** $X \sim Exp(\mu)$ or $X \sim Exponential(\mu)$

### Important Properties for the Exponential Distribution

If $X \sim Exp(\mu)$

- Tail Probability formula: $P(X > x) = e^{-\frac{x}{\mu}}$
- Memoryless Property: $P(X > s + t| X > s) = P(X > t)$

### Example 2

It is your birthday and you are waiting for someone to write you a birthday message on Facebook.
On average (on your birthday) you receive a facebook message every 10 minutes.
Assume that birthday messages arrive independently.

- What is the probability the next posting takes 15 minutes or longer to appear? What distribution, parameter(s) and support are you using?
- What is the standard deviation of the time between birthday postings?
- What is the probability that it takes 12.5 minutes for the next birthday posting?
- Suppose that the most recent birthday posting was done at 1:40 pm and it is now 1:45 pm.
What is the probability that you will have to wait until 1:53 pm or later for the next message?
- What is the probability that your wait time for the next three messages is less than 8 minutes?
- What is your median waiting time for birthday messages?