Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

The relationship between the variance of a sum of two random variables and the concepts of covariance and correlation. It provides formulas and examples to help understand how these statistical measures are calculated and interpreted, and discusses the importance of covariance in the context of investment diversification.

Typology: Exercises

2021/2022

1 / 2

Download Understanding Covariance and Correlation: An Essential Concept in Random Variability and more Exercises Algebra in PDF only on Docsity! Random Variability: Covariance and Correlation What of the variance of the sum of two random variables? If you work through the algebra, you'll find that Var[X+Y] = Var[X] + Var[Y]+ 2(E[XY] - E[X]E[Y]) . This means that variances add when the random variables are independent, but not necessarily in other cases. The covariance of two random variables is Cov[X,Y] = E[ (X-E[X])(Y-E[Y]) ] = E[XY] - E[X]E[Y]. We can restate the previous equation as Var[X+Y] = Var[X] + Var[Y] + 2Cov[X,Y] . Note that the covariance of a random variable with itself is just the variance of that random variable. While variance is usually easier to work with when doing computations, it is somewhat difficult to interpret because it is expressed in squared units. For this reason, the standard deviation of a random variable is defined as the square-root of its variance. A practical (although not quite precise) interpretation is that the standard deviation of X indicates roughly how far from E[X] you’d expect the actual value of X to be. Similarly, covariance is frequently “de-scaled,” yielding the correlation between two random variables: Corr(X,Y) = Cov[X,Y] / ( StdDev(X) StdDev(Y) ) . The correlation between two random variables will always lie between -1 and 1, and is a measure of the strength of the linear relationship between the two variables. Example: Let X be the percentage change in value of investment A in the course of one year (i.e., the annual rate of return on A), and let Y be the percentage change in value of investment B. Assume that you have $1 to invest, and you decide to put a dollars into investment A, and 1- a dollars into B. Then your return on investment from your portfolio will be aX+(1-a)Y, your expected return on investment will be aE[X] + (1-a)E[Y] , and the variance in your return on investment (a measure of the risk inherent in your portfolio) will be a2Var[X] + (1-a)2Var[Y] + 2a(1-a)Cov[X,Y] .