Chapter 8: Probability Mass Functions (pmf’s) and Cumulative Distribution Functions (cdf’s)

Meike Niederhausen and Nicky Wakim

2024-10-14

Learning Objectives

  1. Calculate probabilities for discrete random variables

  2. Calculate and graph a probability mass function (pmf)

  3. Calculate and graph a cumulative distribution function (CDF)

Where are we?

What is a probability mass function?

Definition: probability distribution or probability mass function (pmf)

The probability distribution or probability mass function (pmf) of a discrete r.v. \(X\) is defined for every number \(x\) by \[p_X(x) = \mathbb{P}(X=x) = \mathbb{P}(\mathrm{all }\ \omega\in S:X(\omega) = x)\]

Let’s demonstrate this definition with our coin toss

Example 1

Suppose we toss 3 coins with probability of tails \(p\). If \(X\) is the random variable counting the number of tails, what are the probabilities of each value of \(X\)?

Remarks on the pmf

Properties of pmf

A pmf \(p_X(x)\) must satisfy the following properties:

  • \(0 \leq p_X(x) \leq 1\) for all \(x\).

  • \(\sum \limits_{\{all\ x\}}p_X(x)=1\).

  • Some distributions depend on parameters

    • Each value of a parameter gives a different pmf

    • In previous example, the number of coins tossed was a parameter

      • We tossed 3 coins

      • If we tossed 4 coins, we’d get a different pmf!

    • The collection of all pmf’s for different values of the parameters is called a family of pmf’s

Binomial family of RVs

Example 2

Suppose you toss \(n\) coins, each with probability of tails \(p\). If \(X\) is the number of tails, what is the pmf of \(X\)?

Bernoulli family of RVs

Example 3

Suppose you toss 1 coin, with probability of tails \(p\). If \(X\) is the number of tails, what is the pmf of \(X\)?

Household size (1/5)

Example 4

The table below shows household sizes in 2019. Data are from the U.S. Census.

Size 1 2 3 4 5 or more
Percent 28% 35% 15% 13% 9%

  1. What is the sample space for household sizes?

  2. Define the random variable for household sizes.

  3. Do the values in the table create a pmf? Why or why not?

  4. Make a plot of the pmf.

  5. Write the cdf as a function.

  6. Graph the cdf of household sizes in 2019.

Household size (2/5)

Example 4

The table below shows household sizes in 2019. Data are from the U.S. Census.

Size 1 2 3 4 5 or more
Percent 28% 35% 15% 13% 9%

  1. What is the sample space for household sizes?

  2. Define the random variable for household sizes.

Household size (3/5)

Example 4

The table below shows household sizes in 2019. Data are from the U.S. Census.

Size 1 2 3 4 5 or more
Percent 28% 35% 15% 13% 9%

  1. Do the values in the table create a pmf? Why or why not?

  2. Make a plot of the pmf

What is a cumulative distribution function?

Definition: cumulative distribution function (CDF)

The cumulative distribution function (cdf) of a discrete r.v. \(X\) with pmf \(p_X(x)\), is defined for every value \(x\) by \[F_X(x) = \mathbb{P}(X \leq x) = \sum \limits_{\{all\ y:\ y\leq x\}}p_X(y)\]

Household size (4/5)

Example 4

The table below shows household sizes in 2019. Data are from the U.S. Census.

Size 1 2 3 4 5 or more
Percent 28% 35% 15% 13% 9%
  1. Write the cdf as a function.

Household size (5/5)

Example 4

The table below shows household sizes in 2019. Data are from the U.S. Census.

Size 1 2 3 4 5 or more
Percent 28% 35% 15% 13% 9%
  1. Graph the cdf of household sizes in 2019.

Properties of discrete CDFs

  • \(F(x)\) is increasing or flat (never decreasing)

  • \(\min\limits_x F(x) = 0\)

  • \(\max\limits_xF(x)=1\)

  • CDF is a step function

Chapter 8 Slides