## Introduction to Probability Models

Lecture 11

Qi Wang, Department of Statistics

Sep 14, 2018

## Some Concepts

**Variable:** a variable is an alphabetic character representing a number,
called the value of the variable, which is either arbitrary, not fully specified or unknown

**Quantitative:**
Variable that can be expressed as a number, or quantified

**Qualitative:**
Variable that can't be expressed as a number, or quantified

### Examples

- The age of your car. (Quantitative.)
- The number of hairs on your knuckle. (Quantitative.)
- The softness of a cat. (Qualitative.)
- The color of the sky. (Qualitative.)
- The number of pennies in your pocket. (Quantitative.)

### Random Variable

**Definition:**the value obtained from an experiment has an associated probability
- It is usually abbreviated as RV
**Discrete Random Variable:** coutable number of values
**Continuous Random Variable:**can take on any value in a range

### Probability Mass Function

**Definition:**a function that gives the probability that a
**discrete** random variable is exactly equal to some value.
- It is usually abbreviated as PMF

### Example 1

Flip a fair coin 3 times, let X = the number of heads

- Write out the PMF for X.
- If the coin is no longer fair and P(H) = .7, write out the PMF.

### Some properties of the PMF

- For every x, $0 \le p_X(x) \le 1$
- $\sum_x{p_X(x)} = 1$

### Example 2

$X \sim p_X(x) = P(X = x) = k(5 - x), x \in \{0, 1, 2, 3, 4\}$

- Find the value of k that makes $p_X(x)$ a legitimate/valid probability model
- Find $P(1\le X\le 3)$
- Find $P(X<3|X\ne 0)$
- Find $P(2\le X\le 4 | 0 < X < 4)$