Introduction to Probability Models

Lecture 11

Qi Wang, Department of Statistics

Sep 14, 2018

Random Variables

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


  • 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

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

Some properties of the PMF

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

Example 2

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

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