Chapter 8: Probability Mass Functions (pmf’s) and Cumulative Distribution Functions (cdf’s)
Learning Objectives
Calculate probabilities for discrete random variables
Calculate and graph a probability mass function (pmf)
Calculate and graph a cumulative distribution function (CDF)
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% |
What is the sample space for household sizes?
Define the random variable for household sizes.
Do the values in the table create a pmf? Why or why not?
Make a plot of the pmf.
Write the cdf as a function.
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% |
What is the sample space for household sizes?
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% |
Do the values in the table create a pmf? Why or why not?
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% |
- 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% |
- 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