Chapter 25: Joint densities

Meike Niederhausen and Nicky Wakim

2023-11-08

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)

Mini Lesson Example 1

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

Double Integrals Mini Lesson (2/3)

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. The solution is available in Meike’s video!

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 (Join 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 \[\begin{aligned} f_X(x)&=& \int_{-\infty}^{\infty} f_{X,Y}(x,y)dy\\ f_Y(y)&=& \int_{-\infty}^{\infty} f_{X,Y}(x,y)dx \end{aligned}\]

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 \(0 \leq x \leq 2, \ 0 \leq y \leq 1\).

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

Example of a more complicated joint pdf

Do this problem at home for extra practice. The solution is available in Meike’s video!

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. The solution is available in Meike’s video!

Example 2.2

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

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

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. 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. The solution is available in Meike’s video!

Example 3.3

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

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

Let’s complicate this even further!

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 \(Z=XY\).

Chapter 25 Slides