Correlation – Proofs

On this webpage, we provide proofs for the properties described in Basic Concepts of Correlation.

Property 4: For constant a and random variables x, y, and z, the following are true both for the sample and population definitions of covariance

1. cov(x, y) = cov(y, x)
2. cov(x, x) = var(x)
3. cov(a, y) = 0
4. cov(ax, y) = a · cov(x, y)
5. cov(x+z, y) = cov(x, y) + cov(z, y)

Proof: We give the proofs for the population version of covariance. The proofs for the sample versions are similar.

a)      Clear from the definition of covariance

b)      Clear from the definitions of covariance and variance

c)      Follows from

d)      Follows from

e)      Follows from

Property A: If x and y are random variables and z = ax + b where a and b are constants then the correlation coefficient between x and y is the same as the correlation coefficient between z and y.

Proof: By Property 4

By Property 3 of Expectation

and so stdev(z) = a · stdev(x). Thus

Property 1

Proof: Let

Thus

Now

Collecting identical terms, together with a change in summation indices, we find:

Thus

and so

from which the result follows.

Property 2:

Proof: The proof is similar to the proof of Property 1.

Property 3:

Proof: The proof is similar to that of Property 2 of Expectation.

Property 5: The following is true for both the sample and population definitions of covariance:

If x and y are independent then cov(x, y) = 0

Proof: The proof follows from Property 1d of Expectation and Property 3.

Property 6: The following are true both for samples and populations:

Proof: We give the proof of the first property for the population. The proof of the sample case is similar, as is the proof of the second property. By Property 2 of Expectation and Property 3