## 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.