Covariance
Covariance is a measure of the tendencies of two variables to grow together (positive covariance) or inversely
(negative covariance). Note the similarity to the variance calculation, especially in the short cut.
σXY = Cov(X, Y ) = E[(X−µX) (Y−µY)] = E[XY ]−E[X]E[Y]
If two variables are independent, their covariance is zero. However, a covariance of zero does not guarantee
independence.
Var(aX +bY ) = a2Var(X) + b2Var(Y)+2abCov(X, Y )
Cov(X, Y ) = Cov(Y, X )
Cov(aX, bY ) = abCov(X, Y )
Cov(X, aY +bZ) = aCov(X, Y ) + bCov(X, Z)
Correlation is a unit-free measure of the linear relationshiip between two variables. It is sometimes noted
with either ρor ρXY , and −1≤ρ≤1. Correlation of ±1 implies the variables are linearly related.
Corr(X, Y ) = Cov(X, Y )
σXσY
Many questions involve a sum or product of similar random variables. These questions use different formu-
las, but it can be difficult to understand the distinction. Below is an excerpt from the a/s/m Exam P Study
Manual on the topic.
•Formula for the variance of a sum of independent random variables
VarXXi=XVar (Xi)
•Formula for the variance of constant times variable
Var(nX ) = n2Var(X)
If the variance of a stock is 5 and you buy 10 shares of it, then the variance of your investment is 500.
But if you buy ten independent stocks, each with variance of 5, then the variance of your investment is 50.
See the difference? In the first case, you did not diversify your investment. Any movement of the stock is
magnified by 10. In the second case, you diversified your investment. The movement of one stock does not
get magnified, and in fact may be offset by the movement in the other direction of another stock.
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