Search
STATLECT > Fundamentals of probability > Covariance matrix

Covariance matrix - Exercise set 1

This exercise set contains some solved exercises on covariance matrices. The theory needed to solve these exercises is introduced in the lecture entitled Covariance matrix.

Exercise 1.1

Let X be a $2 imes 1$ random vector and denote its components by X_1 and X_2. The covariance matrix of X is:[eq1]Compute the variance of the random variable Y defined as:[eq2]

nav_button Solution

Using a matrix notation, Y can be written as:[eq3]where we have defined:[eq4]Therefore, the variance of Y can be computed using the formula for the covariance matrix of a linear transformation:[eq5]

Exercise 1.2

Let X be a $3 imes 1$ random vector and denote its components by X_1, X_2 and $X_{3}$. The covariance matrix of X is:[eq6]Compute the following covariance:[eq7]

nav_button Solution

Using the bilinearity of the covariance operator, we obtain:[eq8]The same result can be obtained using the formula for the covariance between two linear transformations. Defining[eq9]we have:[eq10]

Exercise 1.3

Let X be a Kx1 random vector whose covariance matrix is equal to the identity matrix:[eq11]Define a new random vector Y as follows:[eq12]where A is a $K imes K$ matrix of constants such that:[eq13]Derive the covariance matrix of Y.

nav_button Solution

Using the formula for the covariance matrix of a linear transformation:[eq14]

by
About | Contacts | Privacy and terms of use | Sitemap