3.16 Covariance The covariance of any two random variables X and Y, denoted by c

Question

3.16 Covariance The covariance of any two random variables X and Y, denoted by cov(X,Y) is defined by cov(X, Y) = E (X-E [X] )(Y-E [Y])]. (a) Prove that cov (X, Y) = E [XY]-E [X] E [Y]. (b) If X Y, what can we say about cov( X, Y)? (c) Let X and Y be indicator random variables, where X- 1 if event A occurs 0 o.w 3.16 EXERCISES 67 and if event B occurs y. {0 1 B : Y= Prove that if events A and B are positively correlated (see Exercise 3.15), then cov(X, Y) > 0, whereas if A and B are negatively correlated, then cov(X, Y)0. Note: This notion can be extended to general random variables X and Y, not just indicator random variables.

Explanation / Answer

A) Cov(x,y) = E[(x - E(x) (y - E(y)]

= E[xy - x*E(y) - y*E(x) + E(x)*E(y)]

= E(xy) - E(x)E(y)  - E(x)E(y) +  E(x)E(y)

= E(xy) -  E(x)E(y)

Hence proved

b) If x is perpendicular to Y, then both the variables are independent and uncorrelated

if there is no correlation then cov(x, y) is equal to '0'

c) The signs of slope, correlation and covariance are same or equal

so r > 0, then cov(x,y) > 0

r < 0, the cov(x,y) < 0

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