How to Get Marginal CDF from a Joint CDF
In probability theory and statistics, the joint cumulative distribution function (CDF) is a fundamental concept that describes the probability distribution of two or more random variables. However, in many practical applications, we are interested in the marginal distribution of a single random variable. This article aims to provide a comprehensive guide on how to obtain the marginal cumulative distribution function (CDF) from a joint CDF.
The joint CDF of two random variables, X and Y, is defined as:
F(x, y) = P(X ≤ x, Y ≤ y)
where P denotes the probability. The joint CDF can be extended to more than two random variables. In this article, we will focus on the two-dimensional case.
To derive the marginal CDF of a single random variable, say X, from the joint CDF, we need to integrate the joint CDF over the range of the other random variable, Y. The marginal CDF of X, denoted as Fx(x), can be obtained as follows:
Fx(x) = ∫ F(x, y) dy
Here, the integral is taken over the entire range of Y. In practice, this may involve differentiating the joint CDF with respect to y and evaluating the resulting expression at y = x.
Let’s consider an example to illustrate the process. Suppose we have two random variables, X and Y, with the following joint CDF:
F(x, y) = { 0, if x < 0 or y < 0 { x + y, if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 { 1, if x > 1 or y > 1 }
To find the marginal CDF of X, we integrate the joint CDF over the range of Y:
Fx(x) = ∫ F(x, y) dy
For 0 ≤ x ≤ 1, we have:
Fx(x) = ∫_0^1 (x + y) dy
= [xy + (1/2)y^2]_0^1
= x + (1/2)
For x > 1, we have:
Fx(x) = ∫_0^1 (x + y) dy
= [xy + (1/2)y^2]_0^1
= x + (1/2)
For x < 0, we have:
Fx(x) = ∫_0^1 (x + y) dy
= [xy + (1/2)y^2]_0^1
= 0
Therefore, the marginal CDF of X is:
Fx(x) = { x + (1/2), if 0 ≤ x ≤ 1
{ 1, if x > 1
{ 0, if x < 0 }
In conclusion, obtaining the marginal CDF from a joint CDF involves integrating the joint CDF over the range of the other random variable. This process can be applied to any joint CDF, regardless of the number of random variables involved.
