Lecture 14

Joint Distributions

Grace Tompkins

Last modified — 13 Nov 2026

Learning Outcomes

By the end of this lecture, students are anticipated to be able to:

• Calculate and utilize joint distributions

1 Joint Distributions

Joint Distributions

• Suppose \(X\) and \(Y\) are two random variables.
• We may be interested in their distributions separately.
  
• But this ignores the relationship between \(X\) and \(Y\).

The joint CDF of the two random variables \(X\) and \(Y\) is defined by \[F_{X, Y} ( a, b ) = \mathbb{P}( X \le a, \, \ Y \le b).\]

Recall: \[\left\{ X \le a \, , \ Y \le b \right\} = \left\{ X \le a \right\} \cap \left\{ Y \le b \right\}.\]

Joint PMFs and PDFs

Let \(X\) and \(Y\) be two random variables with joint CDF \(F_{X, Y}(x, y)\).

If \(X\) and \(Y\) are both discrete, then the joint PMF of \((X, Y)\) is defined by \[p_{X, Y} ( a, b ) = \mathbb{P}( X = a, \, Y = b).\]

The random variables \(X\) and \(Y\) are jointly (absolutely) continuous if there exists a density function \(f_{X, Y}(x, y)\) such that for any set \(A \subset \mathbb{R}^2\), we have \[\mathbb{P}\left( (X, Y) \in A \right) = \iint_A f_{X, Y}(x, y) \mathsf{d}x \mathsf{d}y.\]

Joint PMF

• Consider the experiment of rolling two fair dice.
• Let \(X\) be the lowest of the two rolls, \(Y\) be the highest.

Find the joint PMF of \(X\) and \(Y\).

This will involve finding all of the joint probabilities for each combination of the dice:

\(f_{X,Y}(x, y)\)	1	2	3	4	5	6
1	1/36	2/36	2/36	2/36	2/36	2/36
2	0	1/36	2/36	2/36	2/36	2/36
3	0	0	1/36	2/36	2/36	2/36
4	0	0	0	1/36	2/36	2/36
5	0	0	0	0	1/36	2/36
6	0	0	0	0	0	1/36

Joint PMF

Suppose \(\mathbb{P}(X=x, Y=y) = c · (x + 2y + 1)I_{\{0,1,2\}}(x)I_{\{0,1\}}(y)\)

1. What value of c will make this a valid joint PMF?

2. What is \(\mathbb{P}(X = 0, Y = 1)\)?

Joint PMF

Joint PDF

Let \(X\) and \(Y\) be jointly continuous, with joint density function

\[ f(x,y) = \begin{cases} 4x^2y + 2y^5 & 0\le x \le 1, 0\le y \le 1\\ 0 &\text{otherwise} \end{cases} \]

1. Verify that this is a density function.

2. What is \(\mathbb{P}(0 \le X \le 0.5, 0 \le y \le 0.75)\)?

Joint PDF

Joint PDF

The joint PDF is a surface:

\[ f(x,y) = \begin{cases} 4x^2y + 2y^5 & 0\le x \le 1, 0\le y \le 1\\ 0 &\text{otherwise} \end{cases} \]

Bivariate Normal distribution

Bivariate Normal Distribution: Given \(\mu _1, \mu_2, \sigma_1, \sigma_2, \rho \in {\mathbb{R}}\) where \(\sigma_1, \sigma_2 > 0\) and \(-1\le \rho \le 1\), the bivariate normal distribution \(\mathcal{N}(\mu,\mu_2, \sigma_1, \sigma_2)\), is given by:

\[\begin{aligned} &f_{X_1, X_2}(x_1, x_2)\\ &= \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1 - \rho^2}} \exp\left\{ -\frac{1}{2(1 - \rho^2)} \left(\left(\frac{x_1-\mu_1}{\sigma_1}\right)^2 - 2\rho \frac{(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2} + \left(\frac{x_2-\mu_2}{\sigma_2}\right)^2 \right) \right\}. \end{aligned}\]

When \(\mu_1 = \mu_2 = \rho = 0\) and \(\sigma_1 = \sigma_2 = 1\), this is the standard bivariate normal distribution.

Joint PDF

Let \(f_{X,Y}(x,y) = e^{-x-y}I_{[0, \infty)}(x)I_{[0, \infty)}(y)\)

What is the probability \(\mathbb{P}(X < Y)\)?

Joint PDF

Checkpoint 4

You may have:

• HANDWRITTEN Cheat Sheet

• Non-graphing calculator

• Writing materials

Phones and laptops put away, please! You can have 15 minutes.

Upcoming Assessments:

• Checkpoint 5 (Sections 2.7 - 2.8: Joint, Marginal, and Conditional Distributions) is on Friday Nov 6th