4.2 Some rules

The following rules apply for rvs $X$ and $Y$ , and constants $c$ and $d$ .

$E[c] = c$ : A constant is expected to always have the same value.
$E[X + c] = E[X] + c$ : The expected value of adding $c$ to $X$ is the same as the expected value of $X$ , plus $c$ .
$E[cX] = c E[X]$ : the expected value of $c$ lots of $X$ is the same as $c$ times the expected values of $X$ .
$\text{var}[c] = 0$ : Constants (that do not vary) have no variance.
$\text{var}[cX] = c^2 \text{var}[X]$ : This is because the definition of variance is about the expected value of the squared-variable.
$\text{var}[X + c] = \text{var}[X]$ : Adding a constant doesn’t change the amount of variation, just where the variable is centred.
$E[cX + dY] = cE[X] + dE[Y]$ : A linear combination of $X$ and $Y$ .
$\text{var}[cX + dY] = c^2E[X] + d^2E[Y] + 2 cd \text{Cov}[X,Y]$ : The variance of a linear combination depends on how correlated they are.

Most of these make sense just by understanding the meaning of expected values and variances.