Question

Let \(\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)^{\prime}\) be an \(n\) -dimensional random vector, with variancecovariance matrix \((2.6 .11) .\) Show that the ith diagonal entry of \(\operatorname{Cov}(\mathbf{X})\) is \(\sigma_{i}^{2}=\) \(\operatorname{Var}\left(X_{i}\right)\) and that the \((i, j)\) th off diagonal entry is \(\operatorname{cov}\left(X_{i}, X_{j}\right) .\)

Short answer

Each diagonal entry of Covariance matrix gives the variance of the respective variable. Each off-diagonal entry gives the covariance between the pair of different variables.

Step-by-step solution

Understanding Variance and Covariance

The variance of a random variable X is a measure of how much values in the distribution vary on average from the mean, denoted as Var(X). The covariance, on the other hand, is a measure of how much two random variables vary together, denoted as Cov(X,Y).

Understanding Covariance matrix

The covariance matrix of a random vector is a matrix where the (i, j) element is the covariance between the ith and jth elements of the random vector. Likewise, the ith diagonal entry of this matrix is the covariance of the ith element with itself, which is essentially the variance of the ith element, i.e., Var(Xi).

Substantiate the Claims

It we consider a typical element \((i,j)\) of the Covariance matrix, when \(i = j\), it represents the Cov(Xi, Xi), which is essentially Var(Xi), supporting the claim that the diagonal elements represent variance of respective elements. \n When \(i \neq j\), it represents the Cov(Xi, Xj), which is the covariance between the ith and jth elements supporting the claim that the off-diagonal elements represent covariance between respective pairs of different random variables.

Conclude

Therefore, the the ith diagonal entry of Covariance matrix of a random vector under consideration represents Variance of the ith variable and any off diagonal entry \((i, j)\) represent the Covariance between ith and jth variable. These are standard properties of the Covariance matrix and pertain to it's structure and definition.

Key concepts

Variance

Variance is a fundamental concept in statistics and probability theory. It measures the spread or dispersion of a set of values. When dealing with a random variable, variance tells you how much the values deviate from their mean (average) value. This deviation is useful for understanding the variability within a data set or a probability distribution.

The formula for the variance of a random variable, denoted as \( \operatorname{Var}(X) \), is given by:\[\operatorname{Var}(X) = E[(X - \mu)^2]\]Where:

• \(X\) represents the random variable.
• \(\mu\) is the expected value or mean of \(X\).
• \(E\) denotes the expected value operator.

The variance is always non-negative, and the larger its value, the more spread out the numbers in the data set or probability distribution are. It's important to know that variance is measured in squared units of the variable, which is why sometimes standard deviation, the square root of variance, is preferred for interpretation.

Covariance

Covariance is another key statistical concept that provides insights into how two random variables change together. Unlike variance, which measures the variability of a single variable, covariance considers pairs of variables and examines whether the variables tend to be above or below their respective means at the same time.

The formula for the covariance between two random variables \(X\) and \(Y\) is:\[\operatorname{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]\]Where:

• \(\mu_X\) and \(\mu_Y\) are the means of \(X\) and \(Y\) respectively.

If the covariance is positive, it implies that the variables tend to move in the same direction. A negative covariance indicates that they tend to move in opposite directions. If covariance is zero, the variables are said to be uncorrelated. However, it is essential to remember that a zero covariance does not necessarily mean the variables are independent.

Random Vector

A random vector is essentially an ordered collection of random variables. Consider it like a vector that contains multiple random variables instead of just one. If you have a data set with multiple related measurements or features, treating them as a random vector allows you to analyze their dependence and relationships in a unified way.

A typical random vector \( \mathbf{X} \) might be denoted as:\[\mathbf{X} = (X_1, X_2, \ldots, X_n)^{\prime}\]Where each \(X_i\) is a random variable. Random vectors are useful in multivariate statistics and help in understanding the joint behavior of several random variables. By using random vectors, we can extend the idea of covariance and variance to multiple dimensions, analyzing how variables interact with one another.

Variance-Covariance Matrix

The Variance-Covariance Matrix, often called just the covariance matrix, is a critical tool in statistics when working with random vectors. It is a matrix that summarizes all the variances and covariances between pairs of random variables in a random vector. This matrix encapsulates the variability and relationships within the variables of interest.

Here's how the covariance matrix is typically structured. For a random vector \( \mathbf{X} = (X_1, X_2, \ldots, X_n)^{\prime} \), the covariance matrix \( \operatorname{Cov}(\mathbf{X}) \) looks like:\[\begin{bmatrix}\operatorname{Var}(X_1) & \operatorname{Cov}(X_1, X_2) & \cdots & \operatorname{Cov}(X_1, X_n) \\operatorname{Cov}(X_2, X_1) & \operatorname{Var}(X_2) & \cdots & \operatorname{Cov}(X_2, X_n) \\vdots & \vdots & \ddots & \vdots \\operatorname{Cov}(X_n, X_1) & \operatorname{Cov}(X_n, X_2) & \cdots & \operatorname{Var}(X_n)\end{bmatrix}\]Each diagonal entry \(i\) represents the variance of \(X_i\), while off-diagonal entries \((i, j)\) capture the covariance between \(X_i\) and \(X_j\). This matrix is symmetric, meaning that \(\operatorname{Cov}(X_i, X_j) = \operatorname{Cov}(X_j, X_i)\). Understanding this matrix is vital in fields like finance, data science, and machine learning, where identifying relationships among different variables is crucial.