Problem 62 - An Introduction to Mathematical Statistics and Its Applications

Question

(a) For random variables \(X\) and \(Y\), show that $$ \operatorname{Cov}(X+Y, X-Y)=\operatorname{Var}(X)-\operatorname{Var}(Y) $$ (b) Suppose that \(\operatorname{Cov}(X, Y)=0\). Prove that $$ \rho(X+Y, X-Y)=\frac{\operatorname{Var}(X)-\operatorname{Var}(Y)}{\operatorname{Var}(X)+\operatorname{Var}(Y)} $$

Step-by-Step Solution

Answer

After expanding and simplifying the covariance and variance formulas, we can clearly see that the covariance of \(X+Y\) and \(X-Y\) is equal to the variance of \(X\) minus the variance of \(Y\), which proves (a). For part (b), using the fact that the covariance of \(X\) and \(Y\) is zero and using step (a), we can prove the given expression is true.

Step 1: Identify and Expand Covariance

Start by expanding the left hand side of part (a) using the definition of covariance. The covariance of \(X+Y\) and \(X-Y\) can be written as follows \(\operatorname{Cov}(X+Y, X-Y) = E[(X+Y-\mu_{X+Y})(X-Y-\mu_{X-Y})]\).

Step 2: Simplify the Resulting Expression

Now, use the properties of expectations to expand and simplify the equation. The properties of expected values allow us to distribute the expectations over the summations. After distribution, some of the terms can be combined, which results in the formula \(\operatorname{Var}(X)-\operatorname{Var}(Y)\).

Step 3: Simplify Using Covariance Equals Zero

In part (b), it's given that the covariance between variables \(X\) and \(Y\) is equal to zero, i.e., \(Cov(X,Y) = 0\). Now, the correlation coefficient \(\rho(X+Y, X-Y)\) can be expressed as \(\rho(X+Y, X-Y) = \frac{Cov(X+Y, X-Y)}{\sqrt{Var(X+Y)Var(X-Y)}}\). Now substitute the \(Cov(X+Y, X-Y)\) expression from step (a) and simplify it.

Step 4: Simplify to get the result

Solving for the denominator in the correlation equation results in the expression \(Var(X+Y)+Var(X-Y)\), substituting this in the earlier equation gives the given expression \(\frac{\operatorname{Var}(X)-\operatorname{Var}(Y)}{\operatorname{Var}(X)+\operatorname{Var}(Y)}\). Thus, proving it.

Key Concepts

Covariance Calculation

Understanding covariance is fundamental to grasping relationships between two random variables. Simply put, covariance measures the joint variability of two random variables. To calculate covariance, we use the following formula: \( \text{Cov}(X, Y) = E[(X - \text{E}[X])(Y - \text{E}[Y])] \). In this formula, \(E\) represents the expected value, or mean, of the variable. The expected value of a variable, such as \(X\), is denoted as \( \text{E}[X] \).

This equation tells us to first find the products of the differences of each variable from their respective means. We then find the average (expected value) of those products, which gives us the covariance. If the covariance is positive, it indicates that as one variable increases, the other tends to increase as well. A negative covariance suggests the opposite relationship, where one variable tends to decrease as the other increases. Zero covariance implies no linear relationship between the variables. It's important to note, covariance is affected by the scale of the variables, making it difficult to interpret the magnitude of covariance without context.

Variance of Random Variables

Variance provides a measure of the spread or dispersion of a set of data points within a random variable. The variance of a single random variable, say \(X\), is calculated using the formula \( \text{Var}(X) = \text{E}[(X - \text{E}[X])^2] \). This represents the expected value of the squared differences between the random variable and its mean.

The variance offers a snapshot of how much a set of observations deviates from the average. It plays a crucial role not only as a standalone concept but also in calculating covariance where it helps us analyze how two variables move in relation to one another. A higher variance means that the data points are more spread out from the mean. When dealing with multiple random variables, such as in the given exercise \( \text{Cov}(X+Y, X-Y) \), understanding the individual variances of \(X\) and \(Y\) is important for assessing how they will interact with one another.

Properties of Expectations

The properties of expectations play a significant role in statistical calculations. These properties include linearity and the ability to distribute expectation over addition or subtraction of random variables. For example, \( \text{E}[X + Y] = \text{E}[X] + \text{E}[Y] \) and \( \text{E}[aX] = a\text{E}[X] \), where \(a\) is a constant.

These properties are invaluable when simplifying complex expressions as shown in our exercise where the covariance of \(X + Y\) and \(X - Y\) can be expanded and simplified using properties of expectations. This simplification assisted in deriving the relationship between variance and covariance, illustrating the interconnectedness of these statistical concepts. A clear understanding of these properties enables one to manipulate equations efficiently, thus proving various statistical theorems and identities.

Correlation Coefficient

The correlation coefficient, often denoted as \( \rho \), measures the strength and direction of the linear relationship between two variables. It is standardized, meaning it provides a value between -1 and 1, which is independent of the scale of the variables. This scale-free measure makes it suitable for comparing correlations between different pairs of variables. The correlation coefficient is calculated using \( \rho(X, Y) = \frac{\text{Cov}(X, Y)}{\text{Var}(X)\text{Var}(Y)} \).

In the context of our exercise, when \(\text{Cov}(X, Y) = 0\), the correlation coefficient for \(X + Y\) and \(X - Y\) simplifies to the expression given, showing a unique relationship between variances and correlation. By understanding how to compute and interpret the correlation coefficient, students can better understand the dynamics of statistical relationships.