Properties of variance and covariance

(a) If and are independent, then by observing that .

(b) In contrast to the expectation, the variance is not a linear operator. For two random variables and , we have

 (3)

However, if and are independent, by observing that in (), we have

 (4)

In general, when we have a sequence of independent random variables , the property () is extended to

Variance and covariance under linear transformation. Let and be scalars (that is, real-valued constants), and let be a random variable. Then the variance of is given by

Now let , , and be scalars, and let and be random variables. Then similarly the covariance of and can be given by

Generated by MATH GO: 2005-09-15