Continuous joint distribution

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continuous joint distribution describes the probability of interaction between two continuous random variables. Its discrete counterpart is the discrete joint distribution which has a countable number of possible outcomes (e.g., 1, 2, 3…).

Continuous joint distributions can be described by a non-negative, integrable function [1]. The leap from discrete joint distributions to continuous ones is much like the leap from single variable discrete random variable to continuous ones. However, as continuous joint distributions are two dimensional, double integrals (from calculus) are needed instead of sigma notation (Σ) to solve probability problems.

The continuous joint distribution PDF

The continuous joint distribution assigns relative likelihoods, via the likelihood function, to combinations of x and y. The probabilities p(x, y) are not traditional probabilities, in the sense that you can get a probability of, say 99% or 50% or 10%; as this is a continuous distribution, the probability of a specific value is always zero. Probabilities for joint continuous distributions are instead treated as volume problems [2], calculate by a double integral ∫∫.

Continuous joint distributions are formally described by a joint probability density function, much in the same way that random variables are described by a “single” probability density function.

Two random variables X and Y are jointly continuous if there exists an integrable non-negative function fXY: ℝ2 → ℝ such that for any set A ∈ ℝ

Continuous joint distribution
fXY (x,y) is the joint PDF of X and Y.

Example 1: Given the following joint PDF, find the constant c:

joint pdf example

Solution: We are given the x and y bounds (0 to 1), so insert the bounds and the given function (x + cy2) into the double integral and solve (hint: here’s the step by step solution on Symbolab):

A few examples of working with joint probability density functions.


[1] Liu, M. STA 611: Introduction to Mathematical Statistics; Lecture 4: Random Variables and Distributions. Retrieved November 10, 2021 from:

[2] Westfall, P. & Henning, K. (2013). Understanding Advanced Statistical Methods. CRC Press.

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