Learning outcomes After you have read this chapter, you should be able to:l . Define Covariance. 2. Calculate the covariance for the discrete and Continuous Random Variables. . Consider the special cases of covariance. 4. Compute correlation. 3 Introduction So tar we nave studied Joint Probability mass / TTY donation and expected value of some function of random variables. It is also of interested, at times, to know whether two random variables have some sort of relationship or not.
For example, someone may be interested in knowing whether marks obtained by a student is positively affected by number of hours denoted to studying by that student or is negatively affected by hours denoted to watching T. V. If X is marks obtained and Y is umber of hours daily spent on studying, then one is interested in knowing whether X and Y are related. If Yes, positively or negatively (the answer we expect is positive) and if it do affect, how strongly are they related.
For answering the above question, we need to learn statistical techniques of covariance and correlation. This chapter aids is understanding these and solving through these techniques. First section of this chapter covers covariance for discrete and continuous random variables and various theorems and its proofs and corollaries are covered. Second section of this chapter focuses on measuring strength of relationship between X and Y called correlation. 1 .
Covariance When X and Y are two random variables and are not independent then covariance between two random variables X and Y is where OX is mean of variable X and AY is mean of variable Y. (X 0 OX) is deviation of X variable from its mean value is deviation of Y variable from its mean value. So covariance is expected value of deviations of X and Y from its respective mean values. If suppose, X and Y are positively related to each other, then this means that when X attain large value then corresponding Y value also tend to be larger and small values of X correspond to small values of Y.
Then most of the probability mass or density will be associated with either both 4 positive or both negative so the product tends to be positive. Thus for strong positive relationship should be positive. For if there exists strong negative relationship, signs of yielding a negative would be opposite, If they are not related at all, then positive product values would tend to be cancelled out with negative product values, yielding near zero.