Exploring the Possibility of Two Different Random Variables Sharing the Same Joint Cumulative Distribution Function
The question of whether two different random variables can have the same joint cumulative distribution function (CDF) but different marginal CDFs is a fascinating and nuanced topic in probability theory. This question delves into the relationship between joint distributions and their marginal distributions, which are crucial in both theoretical and applied statistics.
Understanding Joint and Marginal Distributions
In probability theory, a joint cumulative distribution function (CDF, denoted as ( F(x, y) )) describes the probability that two random variables ( X ) and ( Y ) are simultaneously less than or equal to some values ( x ) and ( y ), respectively. On the other hand, the marginal CDFs (denoted as ( F_X(x) ) and ( F_Y(y) )) describe the probabilities that each random variable is less than or equal to a specific value without considering the other variable.
The Enigma at Hand
The statement "Is it possible for two different random variables to have the same joint CDF but different marginal CDFs?" is initially perplexing because by definition, the joint CDF uniquely determines the marginals. However, this seemingly paradoxical statement can be illuminated by considering the broader context of joint distributions and their characteristics.
Uniqueness of Marginals from Joint CDF
For a bivariate random variable ( (X, Y) ), the marginals can be derived from the joint CDF as follows:
( F_X(x) F(x, infty) ) ( F_Y(y) F(infty, y) )From this, it is clear that if two random variables have the same joint CDF, their marginals must be identical. However, the question asks if it is possible for the marginals to be different. This apparent contradiction can be resolved by examining how much information the joint CDF actually contains.
Complementary Perspective
The key insight lies in recognizing that while the joint CDF determines the marginals, it does not necessarily determine the joint distribution uniquely. There can be multiple joint distributions that share the same marginal distributions. This is because the joint distribution includes the dependence structure between the random variables, which is not captured by the marginals alone.
Example of Different Joint Distributions with Identical Marginals
A classic example of this phenomenon is the case of dependent and independent random variables. Consider two random variables ( X ) and ( Y ) that are independent. Their joint CDF is simply the product of their marginal CDFs:
[F_{X,Y}(x, y) F_X(x) cdot F_Y(y)]However, consider two different random variables with the same marginal distributions but a different dependence structure. For instance, let's say ( U ) and ( V ) are both uniformly distributed between 0 and 1. If ( U ) and ( V ) are independent, their joint CDF is:
[F_{U,V}(u, v) u cdot v]Now, suppose ( W ) and ( Z ) are also uniformly distributed but are positively correlated, such that ( W geq Z ). The exact form of the joint CDF for ( W ) and ( Z ) would be different, but their marginal distributions remain the same.
Using Copulas for Dependent Variables
A more advanced tool to construct different joint distributions with the same marginals is the concept of copulas. A copula is a function that links the marginals to create a joint distribution. Two different copulas can yield the same marginals but different joint distributions. This approach generalizes the idea of independent random variables and allows for a wide range of dependence structures.
Conclusion
It is indeed possible for two different random variables to have the same joint cumulative distribution function but different marginal ones. This possibility arises because the joint CDF contains more information than just the marginals. The marginals only capture the individual distributions of the variables, while the joint CDF includes additional information about their dependence. By exploring the use of copulas, we can construct various joint distributions that share the same marginals but have distinct dependence structures.
Understanding this relationship between joint and marginal distributions is crucial for various applications in statistics and probability theory, including risk management, machine learning, and data analysis.