Problem 45 - A First Course in Probability

Question

If \(X_{1}, X_{2}, X_{3},\) and \(X_{4}\) are (pairwise) uncorrelated random variables, each having mean 0 and variance \(1,\) compute the correlations of (a) \(X_{1}+X_{2}\) and \(X_{2}+X_{3}\) (b) \(X_{1}+X_{2}\) and \(X_{3}+X_{4}\)

Step-by-Step Solution

Answer

The correlation of \(X_1+X_2\) and \(X_2+X_3\) is \(\frac{1}{2}\), whereas the correlation of \(X_1+X_2\) and \(X_3+X_4\) is \(0\).

Step 1: (a) Covariance Calculation

First, we calculate the covariance for (a): \(\operatorname{Cov}(X_1 + X_2, X_2 + X_3) = \mathbb{E}[(X_1 + X_2) (X_2 + X_3)] - \mathbb{E}[X_1 + X_2] \mathbb{E}[X_2 + X_3]\) Since each of the random variables has mean 0, we get: \(\operatorname{Cov}(X_1 + X_2, X_2 + X_3) = \mathbb{E}[(X_1 + X_2) (X_2 + X_3)]\) Expanding the expectation, we have: \(\operatorname{Cov}(X_1 + X_2, X_2 + X_3) = \mathbb{E}[X_1 X_2 + X_1 X_3 + X_2^2 + X_2 X_3]\) Since all the random variables are pairwise uncorrelated, we have: \(\operatorname{Cov}(X_1 + X_2, X_2 + X_3) = \mathbb{E}[X_1 X_2] + \mathbb{E}[X_1 X_3] + \mathbb{E}[X_2^2] + \mathbb{E}[X_2 X_3]\) Applying the definition for pairwise uncorrelated variables, we obtain: \(\operatorname{Cov}(X_1 + X_2, X_2 + X_3) = 0 + 0 + \mathbb{E}[X_2^2] + 0 = \mathbb{E}[X_2^2]\) Since the variance of \(X_2\) is 1, we have \(\mathbb{E}[X_2^2] = 1\).

Step 2: (a) Correlation Calculation

To find the correlation for (a), we use the following formula: \(\rho(X_1 + X_2, X_2 + X_3) = \frac{\operatorname{Cov}(X_1 + X_2, X_2 + X_3)}{\sqrt{\operatorname{Var}(X_1 + X_2) \operatorname{Var}(X_2 + X_3)}}\) As \(X_1\), \(X_2\), and \(X_3\) are pairwise uncorrelated and their variance is 1, we obtain: \(\rho(X_1 + X_2, X_2 + X_3) = \frac{1}{\sqrt{(1 + 1)(1 + 1)}} = \frac{1}{2}\) Thus, the correlation between \(X_1 + X_2\) and \(X_2 + X_3\) is \(\frac{1}{2}\).

Step 3: (b) Covariance Calculation

Now, we calculate the covariance for (a): \(\operatorname{Cov}(X_1 + X_2, X_3 + X_4) = \mathbb{E}[(X_1 + X_2) (X_3 + X_4)] - \mathbb{E}[X_1 + X_2] \mathbb{E}[X_3 + X_4]\) Since each of the random variables has mean 0, we get: \(\operatorname{Cov}(X_1 + X_2, X_3 + X_4) = \mathbb{E}[(X_1 + X_2) (X_3 + X_4)]\) Expanding the expectation, we have: \(\operatorname{Cov}(X_1 + X_2, X_3 + X_4) = \mathbb{E}[X_1 X_3 + X_1 X_4 + X_2 X_3 + X_2 X_4]\) Since all the random variables are pairwise uncorrelated, we have: \(\operatorname{Cov}(X_1 + X_2, X_3 + X_4) = 0 + 0 + 0 + 0 = 0\)

Step 4: (b) Correlation Calculation

To find the correlation for (b), we use the following formula: \(\rho(X_1 + X_2, X_3 + X_4) = \frac{\operatorname{Cov}(X_1 + X_2, X_3 + X_4)}{\sqrt{\operatorname{Var}(X_1 + X_2) \operatorname{Var}(X_3 + X_4)}}\) As \(X_1\), \(X_2\), \(X_3\) and \(X_4\) are pairwise uncorrelated and their variance is 1, we obtain: \(\rho(X_1 + X_2, X_3 + X_4) = \frac{0}{\sqrt{(1 + 1)(1 + 1)}} = 0\) Thus, the correlation between \(X_1 + X_2\) and \(X_3 + X_4\) is \(0\).

Key Concepts

Uncorrelated Random Variables

Understanding the concept of uncorrelated random variables is fundamental in the study of probability theory. When we say that two random variables, let's call them 'A' and 'B', are uncorrelated, it means that knowing information about 'A' does not give us any information about 'B', and vice versa. In more formal terms, the correlation between 'A' and 'B' is zero. This property is especially important because it simplifies many calculations in statistics and probability.

For example, if we are dealing with two uncorrelated random variables with zero mean, any covariance involving these variables evaluates to zero unless they are the same variable. This principle often arises in exercises related to probability distributions and statistical analysis, such as the calculation of the variance of sums of variables or the evaluation of risk in portfolio theory in finance. In our textbook exercise, the random variables are given as having a mean of zero and being pairwise uncorrelated with variances of one, which leads us to straightforward calculations with reduced computational complexity.

Covariance Calculation

Covariance is a measure of how much two random variables change together. In other words, it indicates the degree to which two variables are linearly associated. The calculation of covariance, denoted as \( \operatorname{Cov}(A, B) \), is typically carried out through the expectation of the product of the random variables minus the product of their expected values: \( \operatorname{Cov}(A, B) = \mathbb{E}[(A - \mathbb{E}[A])(B - \mathbb{E}[B])] \).

However, if both A and B have a mean of zero, then the formula simplifies to \( \operatorname{Cov}(A, B) = \mathbb{E}[AB] \), as is seen in our textbook problem. In the context of the exercise, because we're working with uncorrelated random variables, the covariance further simplifies since the expected value of the product of distinct random variables is zero. This leads to an uncomplicated calculation and allows us to quite easily find the covariance for combinations of the random variables given.

Correlation Coefficient

The correlation coefficient, often symbolized as \( \rho \) or 'r', provides a normalized measure of the strength and direction of the linear relationship between two random variables. Its value ranges from -1 to +1, where +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 implies no linear correlation at all.

The correlation coefficient is calculated by dividing the covariance of the variables by the product of their standard deviations: \( \rho(A, B) = \frac{\operatorname{Cov}(A, B)}{\sqrt{\operatorname{Var}(A) \operatorname{Var}(B)}} \). In the exercise provided, after calculating the covariance, we compute the correlation coefficient for the variables. Due to the random variables being uncorrelated and having equal variances, the denominator becomes the square root of the product of their variances, leading to a simplified expression for correlation. Understanding the relationship between covariance and correlation coefficient is key for interpreting statistical data and conducting correlation analysis in fields ranging from finance to social sciences.