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Covariance and Correlation
Math 217 Probability and Statistics
Prof. D. Joyce, Fall 2014
Covariance. Let X and Y be joint random vari- ables. Their covariance Cov(X, Y ) is defined by
Cov(X, Y ) = E((X − μX )(Y − μY )).
Notice that the variance of X is just the covariance of X with itself
Var(X) = E((X − μX )^2 ) = Cov(X, X)
Analogous to the identity for variance
Var(X) = E(X^2 ) − μ^2 X
there is an identity for covariance
Cov(X) = E(XY ) − μX μY
Here’s the proof:
Cov(X, Y ) = E((X − μX )(Y − μY )) = E(XY − μX Y − XμY + μX μY ) = E(XY ) − μX E(Y ) − E(X)μY + μX μY = E(XY ) − μX μY
Covariance can be positive, zero, or negative. Positive indicates that there’s an overall tendency that when one variable increases, so doe the other, while negative indicates an overall tendency that when one increases the other decreases. If X and Y are independent variables, then their covariance is 0:
Cov(X, Y ) = E(XY ) − μX μY = E(X)E(Y ) − μX μY = 0
The converse, however, is not always true. Cov(X, Y ) can be 0 for variables that are not inde- pendent. For an example where the covariance is 0 but X and Y aren’t independent, let there be three outcomes, (− 1 , 1), (0, −2), and (1, 1), all with the same probability 13. They’re clearly not indepen- dent since the value of X determines the value of Y. Note that μX = 0 and μY = 0, so
Cov(X, Y ) = E((X − μX )(Y − μY )) = E(XY ) = 13 (−1) + 13 0 + 13 1 = 0
We’ve already seen that when X and Y are in- dependent, the variance of their sum is the sum of their variances. There’s a general formula to deal with their sum when they aren’t independent. A covariance term appears in that formula.
Var(X + Y ) = Var(X) + Var(Y ) + 2 Cov(X, Y )
Here’s the proof
Var(X + Y ) = E((X + Y )^2 ) − E(X + Y )^2 = E(X^2 + 2XY + Y 2 ) − (μX + μY )^2 = E(X^2 ) + 2E(XY ) + E(Y 2 ) − μ^2 X − 2 μX μY − μ^2 Y = E(X^2 ) − μ^2 X + 2(E(XY ) − μX μY )
- E(Y 2 ) − μ^2 Y = Var(X) + 2 Cov(X, Y ) + Var(Y )
Bilinearity of covariance. Covariance is linear in each coordinate. That means two things. First, you can pass constants through either coordinate:
Cov(aX, Y ) = a Cov(X, Y ) = Cov(X, aY ).
Second, it preserves sums in each coordinate:
Cov(X 1 + X 2 , Y ) = Cov(X 1 , Y ) + Cov(X 2 , Y )
and
Cov(X, Y 1 + Y 2 ) = Cov(X, Y 1 ) + Cov(X, Y 2 ).
Here’s a proof of the first equation in the first condition:
Cov(aX, Y ) = E((aX − E(aX))(Y − E(Y ))) = E(a(X − E(X))(Y − E(Y ))) = aE((X − E(X))(Y − E(Y ))) = a Cov(X, Y )
The proof of the second condition is also straight- forward.
Correlation. The correlation ρXY of two joint variables X and Y is a normalized version of their covariance. It’s defined by the equation
ρXY =
Cov(X, Y ) σX σY
Note that independent variables have 0 correla- tion as well as 0 covariance. By dividing by the product σX σY of the stan- dard deviations, the correlation becomes bounded between plus and minus 1.
− 1 ≤ ρXY ≤ 1.
There are various ways you can prove that in- equality. Here’s one. We’ll start by proving
0 ≤ Var
X
σX
Y
σY
= 2(1 ± ρXY ).
There are actually two equations there, and we can prove them at the same time. First note the “0 ≤” parts follow from the fact variance is nonnegative. Next use the property proved above about the variance of a sum.
Var
X
σX
Y
σY
= Var
X
σX
±Y
σY
X
σX
±Y
σY
Now use the fact that Var(cX) = c^2 Var(X) to rewrite that as
1 σ X^2
Var(X) +
σ^2 Y
Var(±Y ) + 2 Cov
X
σX
Y
σY
But Var(X) = σ^2 X and Var(−Y ) = Var(Y ) = σ^2 Y , so that equals
2 + 2 Cov
X
σX
±Y
σY
By the bilinearity of covariance, that equals
σxσY
Cov(X, Y ) = 2 ± 2 ρXY )
and we’ve shown that
0 ≤ 2(1 ± ρXY.
Next, divide by 2 move one term to the other side of the inequality to get
∓ρXY ≤ 1 ,
so − 1 ≤ ρXY ≤ 1.
This exercise should remind you of the same kind of thing that goes on in linear algebra. In fact, it is the same thing exactly. Take a set of real-valued random variables, not necessarily inde- pendent. Their linear combinations form a vector space. Their covariance is the inner product (also called the dot product or scalar product) of two vectors in that space.
X · Y = Cov(X, Y )
The norm ‖X‖ of X is the square root of ‖X‖^2 defined by
‖X‖^2 = X · X = Cov(X, X) = V (X) = σ X^2
and, so, the angle θ between X and Y is defined by
cos θ =
X · Y
‖X‖ ‖Y ‖
Cov(X, Y ) σX σY
= ρXY
that is, θ is the arccosine of the correlation ρXY.
Math 217 Home Page at http://math.clarku.edu/~djoyce/ma217/
