## Introduction to Probability Models

Lecture 24

Qi Wang, Department of Statistics

Oct 19, 2018

### Uniform Random Variable

X is a continuous random variable with PDF $f_X(x)$

**The definition of $X$ **: the variable is evenly distributed over an interval
**Support:** $X \in [a, b]$ or $a \le X \le b$
**Parameter:** $a, b$, the end points of the interval
**PDF:** $f_X(x) = \frac{1}{b- a}$, for $a \le x \le b$
**CDF: **
\[
F_X(x)=\left\{
\begin{array}{ll}
0, x < a\\
\frac{x - a}{b - a}, a \le x \le b\\
1, x > b
\end{array}
\right.
\]
**Expected Value:** $E[X] = \frac{a + b}{2}$
**Variance:** $Var(X) = \frac{(b - a)^2}{12}$
**Notation:** $X \sim Unif(a, b)$

### Example 1

Let $X \sim Unif(a = 1, b = 5)$

- State and sketch the PDF
- State and sketch the CDF
- Find $F_X(2), F_X(3.7), P(X > 3), P(X < 4 | X > 1.8)$
- Find the mean and standard deviation of X
- What is the $35_{th}$ percentile?

### Example 2

Eric always arrives at his bus stop at 10:05 am, knowing that the arrival of the bus varies anywhere
from 10:05 am to 10:20 am. Let X be the amount of time (in minutes) that Eric waits for the bus to arrive.

- What is the probability that Eric will have to wait longer than 8 minutes? What distribution and parameters are you using?
- What is the probability the bus will come between 10:12 am and 10:18 am?
- Eric has been keeping a record of his wait times for the bus, what is the 40th
percentile of his wait times? What time is that on the clock?
- If Eric's waiting time is at most 9 minutes, what is the probability that it is under 6 minutes?
- After Eric gets on the bus, he has a 10 minute ride and then a 4 minute walk to his
class. His class begins at 10:30 am. What is the probability that he will be on time for
class?