Calculating Correlation Coefficients For Uncorrelated Variables X, Y, And Z

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In probability theory and statistics, understanding the relationships between random variables is crucial. When variables are uncorrelated, it signifies the absence of a linear relationship between them. This article delves into the scenario where we have three uncorrelated random variables, X, Y, and Z, each with specific expected values and variances. We aim to compute the correlation coefficients between different linear combinations of these variables. Specifically, we will calculate ρ(X - Y, X + Y), ρ(X + Y, Y + Z), and ρ(X, Y + Z). This exploration will provide insights into how linear combinations of uncorrelated variables behave and how their correlations can be derived from basic statistical properties.

Suppose that X, Y, and Z are uncorrelated random variables with the following properties:

  • E[X] = E[Y] = E[Z] = 0
  • E[X2] = E[Y2] = E[Z2] = 1

Our goal is to find the correlation coefficients:

  1. ρ(X - Y, X + Y)
  2. ρ(X + Y, Y + Z)
  3. ρ(X, Y + Z)

Before diving into the calculations, let's review the essential concepts. The correlation coefficient, denoted by ρ(A, B), between two random variables A and B is defined as:

ρ(A, B) = Cov(A, B) / (SD(A) * SD(B))

Where Cov(A, B) is the covariance between A and B, and SD(A) and SD(B) are the standard deviations of A and B, respectively. The covariance is defined as:

Cov(A, B) = E[(A - E[A])(B - E[B])]

Since E[X] = E[Y] = E[Z] = 0, the variances are:

Var(X) = E[X2] - (E[X])2 = 1 - 02 = 1 Var(Y) = E[Y2] - (E[Y])2 = 1 - 02 = 1 Var(Z) = E[Z2] - (E[Z])2 = 1 - 02 = 1

Thus, the standard deviations are SD(X) = SD(Y) = SD(Z) = 1. The fact that X, Y, and Z are uncorrelated implies that:

Cov(X, Y) = E[XY] - E[X]E[Y] = E[XY] = 0 Cov(X, Z) = E[XZ] - E[X]E[Z] = E[XZ] = 0 Cov(Y, Z) = E[YZ] - E[Y]E[Z] = E[YZ] = 0

These properties will be essential in our calculations.

1. Correlation Coefficient ρ(X - Y, X + Y)

First, we need to find the covariance between (X - Y) and (X + Y):

Cov(X - Y, X + Y) = E[(X - Y)(X + Y)] - E[X - Y]E[X + Y]

Since E[X] = E[Y] = 0, we have E[X - Y] = E[X] - E[Y] = 0 and E[X + Y] = E[X] + E[Y] = 0. Thus,

Cov(X - Y, X + Y) = E[(X - Y)(X + Y)] = E[X2 - Y2] = E[X2] - E[Y2] = 1 - 1 = 0

Next, we need to find the standard deviations of (X - Y) and (X + Y):

Var(X - Y) = Var(X) + Var(Y) - 2Cov(X, Y) = 1 + 1 - 2(0) = 2 SD(X - Y) = √2

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y) = 1 + 1 + 2(0) = 2 SD(X + Y) = √2

Now we can calculate the correlation coefficient:

ρ(X - Y, X + Y) = Cov(X - Y, X + Y) / (SD(X - Y) * SD(X + Y)) = 0 / (√2 * √2) = 0

Therefore, the correlation coefficient ρ(X - Y, X + Y) is 0. This indicates that the linear combinations X - Y and X + Y are uncorrelated.

2. Correlation Coefficient ρ(X + Y, Y + Z)

Now, let's compute the correlation coefficient ρ(X + Y, Y + Z). First, we find the covariance:

Cov(X + Y, Y + Z) = E[(X + Y)(Y + Z)] - E[X + Y]E[Y + Z]

Since E[X] = E[Y] = E[Z] = 0, we have E[X + Y] = 0 and E[Y + Z] = 0. Thus,

Cov(X + Y, Y + Z) = E[(X + Y)(Y + Z)] = E[XY + XZ + Y2 + YZ]

Using the linearity of expectation and the fact that E[XY] = E[XZ] = E[YZ] = 0, we get:

Cov(X + Y, Y + Z) = E[XY] + E[XZ] + E[Y2] + E[YZ] = 0 + 0 + 1 + 0 = 1

Next, we need to find the standard deviations of (X + Y) and (Y + Z). We already know that SD(X + Y) = √2. Similarly,

Var(Y + Z) = Var(Y) + Var(Z) + 2Cov(Y, Z) = 1 + 1 + 2(0) = 2 SD(Y + Z) = √2

Now we calculate the correlation coefficient:

ρ(X + Y, Y + Z) = Cov(X + Y, Y + Z) / (SD(X + Y) * SD(Y + Z)) = 1 / (√2 * √2) = 1 / 2

Therefore, the correlation coefficient ρ(X + Y, Y + Z) is 1/2. This indicates a positive correlation between X + Y and Y + Z.

3. Correlation Coefficient ρ(X, Y + Z)

Finally, we compute the correlation coefficient ρ(X, Y + Z). First, we find the covariance:

Cov(X, Y + Z) = E[X(Y + Z)] - E[X]E[Y + Z]

Since E[X] = 0 and E[Y + Z] = 0, we have:

Cov(X, Y + Z) = E[X(Y + Z)] = E[XY + XZ] = E[XY] + E[XZ] = 0 + 0 = 0

Next, we already know SD(X) = 1 and SD(Y + Z) = √2. Now we calculate the correlation coefficient:

ρ(X, Y + Z) = Cov(X, Y + Z) / (SD(X) * SD(Y + Z)) = 0 / (1 * √2) = 0

Therefore, the correlation coefficient ρ(X, Y + Z) is 0. This indicates that X and Y + Z are uncorrelated.

In summary, we have found the following correlation coefficients:

  1. ρ(X - Y, X + Y) = 0
  2. ρ(X + Y, Y + Z) = 1/2
  3. ρ(X, Y + Z) = 0

These results highlight how linear combinations of uncorrelated variables can exhibit different correlation behaviors. While the combinations X - Y and X + Y are uncorrelated, and X and Y + Z are uncorrelated, the combinations X + Y and Y + Z show a positive correlation. This is because Y is a common component in both X + Y and Y + Z, which induces a correlation even though X, Y, and Z are individually uncorrelated.

The principles demonstrated in this problem have broad applications in various fields such as finance, engineering, and data science. In finance, understanding the correlation between different assets is crucial for portfolio diversification. By combining assets with low or negative correlations, investors can reduce the overall risk of their portfolio. In signal processing, uncorrelated signals are often desirable as they represent independent sources of information. In machine learning, feature decorrelation techniques are used to reduce redundancy in the dataset, which can improve the performance of models. The calculations and results presented here provide a foundational understanding of how to analyze and interpret the relationships between random variables and their linear combinations, making it a valuable tool in many analytical contexts.

This article has explored the correlation coefficients between linear combinations of uncorrelated random variables X, Y, and Z. By applying the definitions of covariance and correlation, we calculated ρ(X - Y, X + Y) = 0, ρ(X + Y, Y + Z) = 1/2, and ρ(X, Y + Z) = 0. These findings illustrate the importance of understanding how linear combinations of variables can induce correlations, even when the original variables are uncorrelated. The concepts and techniques discussed are fundamental in probability theory and statistics, and they have wide-ranging applications in various domains. A solid grasp of these principles enables more effective analysis and decision-making in diverse real-world scenarios.

By understanding these relationships, professionals can make more informed decisions and develop more robust strategies in their respective fields. The insights gained from this analysis can be used to optimize portfolio diversification, improve signal processing techniques, and enhance machine-learning models, among other applications. As the world becomes increasingly data-driven, the ability to analyze and interpret statistical relationships will continue to be a valuable asset for professionals across various disciplines.