Properties of variance and covariance

(a) If $ X$ and $ Y$ are independent, then $ {\rm Cov}(X,Y) = 0$ by observing that $ E[XY] = E[X] \cdot E[Y]$.

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

$\displaystyle {\rm Var}(X + Y) = {\rm Var}(X) + {\rm Var}(Y) + 2 {\rm Cov}(X,Y).$ (3)

However, if $ X$ and $ Y$ are independent, by observing that $ {\rm Cov}(X,Y) = 0$ in (*), we have

$\displaystyle {\rm Var}(X + Y) = {\rm Var}(X) + {\rm Var}(Y).$ (4)

In general, when we have a sequence of independent random variables $ X_1,\ldots,X_n$, the property (*) is extended to

$\displaystyle {\rm Var}(X_1 + \cdots + X_n)
= {\rm Var}(X_1) + \cdots + {\rm Var}(X_n).
$

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

$\displaystyle {\rm Var}(aX + b) = a^2 {\rm Var}(X).
$

Now let $ a_1$, $ a_2$, $ b_1$ and $ b_2$ be scalars, and let $ X$ and $ Y$ be random variables. Then similarly the covariance of $ a_1 X + b_1$ and $ a_2 Y + b_2$ can be given by

$\displaystyle {\rm Cov}(a_1 X + b_1, a_2 Y + b_2) = a_1 a_2 {\rm Cov}(X,Y)
$



Generated by MATH GO: 2005-09-15