aX+b (Edexcel IAL Maths)

Spec point — 5.3

aX+b

How are the mean and variance of X related to the mean and variance of aX + b?

• If a and b are constants then the following results are true

  • E(aX + b) = aE(X) + b **
  • Var(aX + b) = a² Var(X)**

• Note that the mean is affected by multiplication and addition whereas addition does not change the variance

• The factor of a² includes the squared because the values of X are squared in the calculation

  • You could try and use the first result and the formula for variance to verify the second result

• Remember a subtraction can be written as an addition

  • X – b can be written as X + (-b)

• And division can be written as a multiplication

  • $\frac{x}{a}$can be written as $\frac{1}{a}X$

Worked Example $X$ is a random variable such that $E(X)=5$ and $\mathrm{Var}(X)=4$.

Find the value of:

(i) $E(3X+5)$

(ii) $\mathrm{Var}(3X+5)$

(iii) $\mathrm{Var}(2-X)$

Answer: