aX+b (Edexcel International AS Maths: Statistics 1): Revision Note

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 $\frac{1}{a}X$