< Probability distributions list
A continuous joint distribution can be described by a non-negative, integrable function [1]. It 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…).
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 to combinations (x,y). The numbers p(x,y) are not 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.
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
Probabilities for joint continuous distributions are volume problems [2], represented by a double integral ∫ ∫.
Example 1: Given the following joint PDF, find the constant c:
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):
References
[1] Liu, M. STA 611: Introduction to Mathematical Statistics; Lecture 4: Random Variables and Distributions. Retrieved November 10, 2021 from: http://www2.stat.duke.edu/courses/Fall18/sta611.01/Lecture/Lecture04.pdf
[2] Westfall, P. & Henning, K. (2013). Understanding Advanced Statistical Methods. CRC Press.