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.