Question

Here are a few useful relationships related to the covariance of two random variables, \(x_{1}\) and \(x_{2}\) a show that \(\operatorname{Cov}\left(x_{1}, x_{2}\right)=E\left(x_{1} x_{2}\right)-E\left(x_{1}\right) E\left(x_{2}\right) .\) An important implication of this is that if \(\operatorname{Cov}\left(x_{1}, x_{2}\right)=0, E\left(x_{1} x_{2}\right)\) \(=E\left(x_{1}\right) E\left(x_{2}\right) .\) That is, the expected value of a product of two random variables is the product of these variables' expected values. b. Show that \(\operatorname{Var}\left(a x_{1}+b x_{2}\right)=a^{2} \operatorname{Var}\left(x_{1}\right)+b^{2} \operatorname{Var}\left(x_{2}\right)+2 a b \operatorname{Cov}\left(x_{1}, x_{2}\right)\) c. In Problem \(2.15 \mathrm{d}\) we looked at the variance of \(X=k x_{1}+(1-k) x_{2} \quad 0 \leq k \leq 1 .\) Is the conclusion that this variance is minimized for \(k=0.5\) changed by considering cases where \(\operatorname{Cov}\left(x_{1}, x_{2}\right) \neq 0 ?\) d. The correlation coefficient between two random variables is defined as \\[ \operatorname{Corr}\left(x_{1}, x_{2}\right)=\frac{\operatorname{Cov}\left(x_{1}, x_{2}\right)}{\sqrt{\operatorname{Var}\left(x_{1}\right) \operatorname{Var}\left(x_{2}\right)}} \\] Explain why \(-1 \leq \operatorname{Corr}\left(x_{1}, x_{2}\right) \leq 1\) and provide some intuition for this result. e. Suppose that the random variable \(y\) is related to the random variable \(x\) by the linear equation \(y=\alpha+\beta x\). Show that \\[ \beta=\frac{\operatorname{Cov}(y, x)}{\operatorname{Var}(x)} \\] Here \(\beta\) is sometimes called the (theoretical) regression coefficient of \(y\) on \(x\). With actual data, the sample analog of this expression is the ordinary least squares (OLS) regression coefficient.

Short answer

In part (a) of the solution, the covariance formula is expanded to show that if the covariance between two random variables is zero, then the expected value of their product equals the product of their expected values. This implies that the two variables are uncorrelated and do not linearly depend on each other. In part (b), the formula for the variance of a linear combination of two random variables is proven. This formula shows the relationship between the variances and covariance of the individual variables. Part (c) analyzes the impact of non-zero covariance on variance minimization. The conclusion is that the minimum-variance value of k may change depending on the specific values of the covariance and variances of the variables involved. Part (d) discusses the correlation coefficient and its bounds between -1 and 1. The bounds represent the strongest possible linear relationships between two variables, with values between -1 and 1 indicating weaker or no linear relationship. Finally, part (e) proves the relationship between the theoretical regression coefficient of y on x, showing that it is equal to the ratio of the covariance between y and x to the variance of x.

Step-by-step solution

Part (a)

The covariance of two random variables \(x_{1}\) and \(x_{2}\) is given by the formula: \[ \operatorname{Cov}\left(x_{1}, x_{2}\right) = E\left[(x_{1} - E\left[x_{1}\right])(x_{2} - E\left[x_{2}\right])\right] \] Expanding the product inside the expectation, we get: \[ \operatorname{Cov}\left(x_{1}, x_{2}\right) = E\left[x_{1} x_{2}\right] - E\left[x_{1}\right] E\left[x_{2}\right] \] This shows that if the covariance between two random variables is zero, then the expected value of their product equals the product of their expected values. In this case, the two variables are uncorrelated and do not linearly depend on each other.

Part (b)

We are asked to prove the following formula: \[ \operatorname{Var}\left(a x_{1}+b x_{2}\right)=a^{2}\operatorname{Var}\left(x_{1}\right)+b^{2}\operatorname{Var}\left(x_{2}\right)+2 a b \operatorname{Cov}\left(x_{1}, x_{2}\right) \] To do this, we first write the variance expression: \[ \operatorname{Var}\left(a x_{1}+b x_{2}\right) = E\left[\left(a x_{1}+b x_{2} - E\left[a x_{1}+b x_{2}\right]\right)^2\right] \] Expanding the square and using the linearity of the expectation operator, we get: \[ \operatorname{Var}\left(a x_{1}+b x_{2}\right) = a^2 E\left[\left(x_{1} - E\left[x_{1}\right]\right)^2\right] + b^2 E\left[\left(x_{2} - E\left[x_{2}\right]\right)^2\right] + 2ab E\left[\left(x_{1} - E\left[x_{1}\right]\right)\left(x_{2} - E\left[x_{2}\right]\right)\right] \] Now, we simply recognize the expressions inside the expectation as the variances and covariance: \[ \operatorname{Var}\left(a x_{1}+b x_{2}\right)=a^{2}\operatorname{Var}\left(x_{1}\right)+b^{2}\operatorname{Var}\left(x_{2}\right)+2 a b \operatorname{Cov}\left(x_{1}, x_{2}\right) \]

Part (c)

In Problem 2.15d, the variance of \(X=k x_{1}+(1-k) x_{2}\) is minimized for \(k=0.5\) when the covariance between \(x_1\) and \(x_2\) is zero. We're asked to check if this conclusion still holds for cases where \(\operatorname{Cov}(x_1, x_2) \neq 0\). Using the formula proven in Part (b), when the covariance is non-zero, choosing a particular value of \(k\) will impact how the covariance term contributes to the overall variance. Therefore, it is possible that the minimum-variance value of \(k\) will change, and the answer to this question depends on the specific values of the covariance and variances of \(x_1\) and \(x_2\).

Part (d)

The correlation coefficient between two random variables is given by: \[ \operatorname{Corr}\left(x_{1}, x_{2}\right) = \frac{\operatorname{Cov}\left(x_{1}, x_{2}\right)}{\sqrt{\operatorname{Var}\left(x_{1}\right) \operatorname{Var}\left(x_{2}\right)}} \] By definition, the variances and the square root we are dividing by are always positive, so the sign of the correlation is determined by the covariance. If the covariance is zero, the correlation will be also zero, meaning there's no linear dependence between the two variables. A correlation of 1 or -1 means the two variables have a perfect positive or negative linear relationship, respectively. Since the covariance is divided by the square root of the product of the variances, the correlation coefficient values are bounded between -1 and 1. These bounds represent the strongest possible linear relationships between the variables, and any value between -1 and 1 indicates weaker or no linear relationship.

Part (e)

We are given the linear equation \(y = \alpha + \beta x\), and are asked to prove that: \[ \beta=\frac{\operatorname{Cov}(y, x)}{\operatorname{Var}(x)} \] First, rewrite the equation as \(y - \alpha = \beta x\). Now calculate the covariance between \(y\) and \(x\): \[ \operatorname{Cov}(y, x) = E\left[(y - E[y])(x - E[x])\right] \] Substitute the equation for \(y\): \[ \operatorname{Cov}(y, x) = E\left[(\beta x - E[\beta x])(x - E[x])\right] \] Simplify by taking the constant \(\beta\) out of the expectations and using the fact that \(E[\beta x] = \beta E[x]\): \[ \operatorname{Cov}(y, x) = \beta E\left[(x - E[x])(x - E[x])\right] \] Recognize the expression inside the expectation as the variance of \(x\): \[ \operatorname{Cov}(y, x) = \beta \operatorname{Var}(x) \] Finally, isolate \(\beta\): \[ \beta=\frac{\operatorname{Cov}(y, x)}{\operatorname{Var}(x)} \]

Key concepts

Variance

Variance is a measure of how much a set of random variables differ from their mean. It gives us an idea of the spread or dispersion within a dataset. Mathematically, for a random variable \(x\), the variance is defined as:\[\operatorname{Var}(x) = E[(x - E[x])^2]\]Here's why variance matters:

• It quantifies the unpredictability or volatility in the dataset.
• A high variance indicates that the data points are spread out over a wider range.
• A low variance indicates that the data points are closer to the mean.

Variance plays a critical role in statistics, especially in formulas like the one for the sum of variables, \(a x_1 + b x_2\), showing how the interrelation of data (via covariance) contributes to variability.

Correlation Coefficient

The correlation coefficient measures the strength and direction of a linear relationship between two random variables, \(x_1\) and \(x_2\). It is expressed as:\[\operatorname{Corr}(x_1, x_2) = \frac{\operatorname{Cov}(x_1, x_2)}{\sqrt{\operatorname{Var}(x_1) \operatorname{Var}(x_2)}}\]Key characteristics of the correlation coefficient include:

• Values range from -1 to 1.
• A value of 1 indicates a perfect positive linear relationship.
• A value of -1 indicates a perfect negative linear relationship.
• A value of 0 means no linear relationship at all.

The correlation is bound by -1 and 1 because it standardizes covariance into a dimensionless value, reflecting the extent to which their movements are synchronized.

Regression Coefficient

In the context of a linear equation, such as \(y = \alpha + \beta x\), the regression coefficient \(\beta\) represents the slope of the line, indicating how much \(y\) is expected to change as \(x\) changes. Mathematically, \(\beta\) is calculated as:\[\beta = \frac{\operatorname{Cov}(y, x)}{\operatorname{Var}(x)}\]Here's more on regression coefficients:

• \(\beta\) tells us the change in \(y\) for a unit increase in \(x\).
• It is derived from the principle of minimizing squared differences (Ordinary Least Squares).
• A positive \(\beta\) suggests \(y\) increases with \(x\), while a negative \(\beta\) suggests the opposite.

Regression coefficients are vital for understanding relationships within data and making predictions.

Expectation

Expectation, or expected value, of a random variable provides a measure of its central tendency. For a discrete random variable \(x\), it is calculated as:\[E[x] = \sum{x_i P(x_i)}\]Whereas for continuous variables, it integrates over a probability density function:\[E[x] = \int{x f(x) \, dx}\]Key elements of expectation include:

• It gives us the average outcome if an experiment is repeated many times.
• Expectation helps in predicting long-term results.
• It facilitates the calculation of higher statistical metrics like variance and covariance.

Expectation is central to probability theory and serves as a foundation for further statistical analysis.