Understanding the Correlation Coefficient Between X and Y When VarXY 0

When dealing with the relationship between two random variables X and Y, one important aspect to consider is the variance of their sum, Var(X Y). In this article, we explore the scenario where Var(X Y) 0. We will delve into the implications of this condition, which plays a crucial role in determining the nature of the relationship between X and Y.

Variance and Independence

The variance of the sum of two random variables X and Y can be expressed using the following formula:

Var(X Y) Var(X) Var(Y) 2Cov(X, Y)

If Var(X Y) 0, it implies that:

The sum X Y is a constant value. Both X and Y must also be constant values with probability 1.

Let's explore these implications in more detail.

Constant Values

If X and Y are constant values, say X a and Y b, the variances of X and Y are:

Var(X) 0 Var(Y) 0 Cov(X, Y) 0

Given these constant values, the correlation coefficient between X and Y, typically denoted as ρ(X, Y), can be regarded as 1 or -1. This is because their values are perfectly correlated or perfectly anticorrelated, depending on the signs of the constants.

Joint Distribution Function

Consider the joint distribution function f_{X,Y} δ_{X-Y} G(y). This function is located on the line y -x, with a variable distribution along this line, where G(y) is the marginal distribution of y. Similarly, G(x) is the marginal distribution of x.

Let's derive the expected values and variances:

The expected value, E[X], is approximately equal to -y, denoted as μ. The expected value of X^2 is approximately equal to y^2, denoted as ω^2.

Therefore, the variance of X is Var(X) E[X^2] - (E[X])^2 ω^2 - μ^2, which is equal to Vary. Substituting these values, we get:

xy -ω^2

x - xy - y μ^2 - ω^2

The correlation coefficient is then:

ρ(X, Y) (x - xy - y) / √(Var(X) * Var(Y)) -1

This result aligns with our intuition, and it holds for any distribution along the line y -x, indicating that G(y) can be any marginal distribution.

Conclusion

The correlation coefficient between X and Y, when Var(X Y) 0, indicates a perfect correlation or antcorrelation, depending on the nature of the random variables. This perfect relationship can manifest in various forms, such as X -Y or when one variable is a constant.

However, it's important to note that the correlation coefficient only captures linear relationships. For non-linear relationships, such as f_{X,Y} G(y)δ_{y-x^2}, the correlation coefficient does not necessarily represent the true relationship between the variables. The concept of linear correlation remains a fundamental tool but should be complemented with other statistical measures to fully understand the relationship between random variables.

In conclusion, the condition Var(X Y) 0 reveals the nature of the relationship between X and Y, providing a valuable insight into their correlation.