Definition of Covariance

1
Definition of Covariance

• The covariance of X Y, denoted Cov(X,Y), is the
  numberwhere mX E(X) and mY E(Y).
• Computational Formula

2
Variance of a Sum
3
Covariance and Independence

• If X Y are independent, then Cov(X,Y) 0.
• If Cov(X,Y) 0, it is not necessarily true that
  X Y are independent!

4
The Sign of Covariance

• If the sign of Cov(X,Y) is positive,
  above-average values of X tend to be associated
  with above-average values of Y and below-average
  values of X tend to be associated with
  below-average values of Y.
• If the sign of Cov(X,Y) is negative,
  above-average values of X tend to be associated
  with below-average values of Y and vice versa.
• If the Cov(X,Y) is zero, no such association
  exists between the variables X and Y.

5
Correlation

• The sign of the covariance has a nice
  interpretation, but its magnitude is more
  difficult to interpret.
• It is easier to interpret the correlation of X
  and Y.
• Correlation is a kind of standardized covariance,
  and

6
Conditions for X Y to be Uncorrelated

• The following conditions are equivalent Corr(X,Y
  ) 0 Cov(X,Y) 0 E(XY) E(X)E(Y)in which
  case X and Y are uncorrelated.
• Independent variables are uncorrelated.
• Uncorrelated variables are not necessarily
  independent!

7

• Let (X, Y) have uniform distribution on the four
  points (-1,0), (0,1), (0,-1) and (1,0). Show that
  X and Y are uncorrelated but not independent.
• What is the variance of X Y?

8

• Let T1 and T3 be the times of the first and third
  arrivals in a Poisson process with rate l.
• Find Corr(T1,T3).
• What is the variance of T1 T3?