Chapter 25: Joint densities

Meike Niederhausen and Nicky Wakim

2024-11-20

Learning Objectives

1. Solve double integrals in our mini lesson!

2. Calculate probabilities for a pair of continuous random variables

3. Calculate a joint and marginal probability density function (pdf)

4. Calculate a joint and marginal cumulative distribution function (CDF) from a pdf

Double Integrals Mini Lesson (1/3)

Do this problem at home for extra practice. I’ll add the solution to the annotated notes!

Mini Lesson Example 1

Solve the following integral: \(\displaystyle\int_{2}^{3}\displaystyle\int_{0}^{1} xy dydx\)

Double Integrals Mini Lesson (2/3)

Do this problem at home for extra practice. I’ll add the solution to the annotated notes!

Mini Lesson Example 2

Solve the following integral: \(\displaystyle\int_{2}^{3}\displaystyle\int_{0}^{1} (x+y) dydx\)

Double Integrals Mini Lesson (3/3)

Do this problem at home for extra practice. I’ll add the solution to the annotated notes!

Mini Lesson Example 3

Solve the following integral: \(\displaystyle\int_{2}^{3}\displaystyle\int_{0}^{1} e^{x+y} dydx\)

How to define the joint pdf for continuous RVs?

For a single continuous RV \(X\) is a function \(f_X(x)\), such that for all real values \(a,b\) with \(a \leq b\), \[\mathbb{P}(a \leq X \leq b) = \int_a^b f_X(x)dx\]

For two continuous RVs (\(X\) and \(Y\)), we can define the joint pdf, \(f_{X,Y}(x,y)\), such that for all real values \(a,b, c, d\) with \(a \leq b\) and \(c \leq d\), \[\mathbb{P}(a \leq X \leq b, c \leq Y \leq d) = \int_a^b \int_c^d f_{X,Y}(x,y)dydx\]

Important properties of the joint pdf

1. Note that \(f_{X,Y}(x,y)\neq \mathbb{P}(X=x, Y=y)\)!!!

2. In order for \(f_{X,Y}(x,y)\) to be a pdf, it needs to satisfy the properties

   • \(f_{X,Y}(x,y)\geq 0\) for all \(x,y\)

   • \(\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty} f_{X,Y}(x,y)dxdy=1\)

What is the joint cumulative distribution function?

Definition: Joint cumulative distribution function (Joint CDF)

The joint cumulative distribution function (cdf) of continuous random variables \(X\) and \(Y\), is the function \(F_{X,Y}(x,y)\), such that for all real values of \(x\) and \(y\), \[F_{X,Y}(x,y)= \mathbb{P}(X \leq x, Y \leq y) = \int_{-\infty}^x\int_{-\infty}^y f_{X,Y}(s,t)dtds\]

Remarks:

• The definition above for \(F_{X,Y}(x,y)\) is a function of \(x\) and \(y\).

• The joint cdf at the point \((a,b)\), is \[F_{X,Y}(a,b) = \mathbb{P}(X \leq a, Y \leq b) = \int_{-\infty}^a\int_{-\infty}^b f_{X,Y}(s,t)dtds\]

What are the marginal pdf’s?

Definition: Marginal pdf’s

Suppose \(X\) and \(Y\) are continuous r.v.’s, with joint pdf \(f_{X,Y}(x,y)\). Then the marginal probability density functions are \[\begin{aligned} f_X(x)&=& \int_{-\infty}^{\infty} f_{X,Y}(x,y)dy\\ f_Y(y)&=& \int_{-\infty}^{\infty} f_{X,Y}(x,y)dx \end{aligned}\]

Common steps for solving problems

1. Set up the domain of the pdf with a picture

2. Translate to needed integrands

   • For probability: shade in the area of interest, then translate
   • For expected value: translate domain

3. Set up integral: \(dxdy\) or \(dydx\)?

4. Solve integral!

Example of joint pdf

Example 1.1

Let \(f_{X,Y}(x,y)= \frac32 y^2\), for \(0 \leq x \leq 2, \ 0 \leq y \leq 1\).

1. Find \(\mathbb{P}(0 \leq X \leq 1, 0 \leq Y \leq \frac12)\).

Example of joint pdf

Example 1.2

Let \(f_{X,Y}(x,y)= \frac32 y^2\), for \(0 \leq x \leq 2, \ 0 \leq y \leq 1\).

2. Find \(f_X(x)\) and \(f_Y(y)\).

Example of a more complicated joint pdf

Do this problem at home for extra practice. I’ll add the solution to the annotated notes!

Example 2.1

Let \(f_{X,Y}(x,y)= 2 e^{-(x+y)}\), for \(0 \leq x \leq y\).

1. Find \(f_X(x)\) and \(f_Y(y)\).

Example of a more complicated joint pdf

Do this problem at home for extra practice. I’ll add the solution to the annotated notes!

Example 2.2

Let \(f_{X,Y}(x,y)= 2 e^{-(x+y)}\), for \(0 \leq x \leq y\).

2. Find \(\mathbb{P}(Y < 3)\).

Let’s complicate this even more!

Example 3.1

Let \(X\) and \(Y\) have constant density on the square \(0 \leq X \leq 4, 0 \leq Y \leq 4\).

1. Find \(\mathbb{P}(|X-Y| < 2)\)

Finding the pdf of a transformation

• Let \(M\) be a transformation of \(X\) and \(Y\)

• When we have a transformation of \(X\) and \(Y\), \(M\), we need to follow a specific process to find the pdf of \(M\)

We follow this process:

1. Start with the joint pdf for \(X\) and \(Y\)

   • aka \(f_{X,Y}(x, y)\)

2. Translate the domain of \(X\) and \(Y\) to \(M\)

3. Find the CDF of \(M\)

   • aka \(F_M(m)\) or \(P(M \leq m)\)

4. Take the derivative of the CDF of \(M\) to find the pdf of \(M\)

   • aka \(f_M(m) = \dfrac{d}{dm}F_M(m)\)

Let’s complicate this even more!

Example 3.2

Let \(X\) and \(Y\) have constant density on the square \(0 \leq X \leq 4, 0 \leq Y \leq 4\).

2. Let \(M = \max(X,Y)\). Find the pdf for \(M\), that is \(f_M(m)\).

Let’s complicate this even more!

Do this problem at home for extra practice. I’ll add the solution to the annotated notes!

Example 3.3

Let \(X\) and \(Y\) have constant density on the square \(0 \leq X \leq 4, 0 \leq Y \leq 4\).

3. Let \(Z = \min(X,Y)\). Find the pdf for \(Z\), that is \(f_Z(z)\).

Let’s complicate this even further!

Do this problem at home for extra practice. I’ll add the solution to the annotated notes!

Example 4

Let \(X\) and \(Y\) have joint density \(f_{X,Y}(x,y)= \frac85(x+y)\) in the region \(0 < x < 1,\ \frac12 < y <1\). Find the pdf of the r.v. \(Z\), where \(Z=XY\).