Linear Transformations of DRVs | Edexcel A Level Further Maths: Further Statistics 1 Revision Notes 2017

E(X) & Var(X) of aX+b

If $a$ and $b$ are constants then $a X + b$ is a linear transformation of $X$
These are two important formulae:
- $E (a X \pm b) = a E (X) \pm b$
- $Var (a X \pm b) = a^{2} Var (X)$
If values of $X$ are multiplied by $a$ , then
- the mean is multiplied by $a$
- the variance is multiplied by $a^{2}$
  - as variance is the square of the spread (standard deviation)
If an amount $b$ is added to $X$
- $b$ is added to the mean
- the variance is unchanged
  - adding $b$ to every value won't make them more spread out
Sometimes it helps to rewrite expressions into the form $a X + b$
- $1 - \frac{X}{4}$ is the same as $- \frac{1}{4} X + 1$

The expectation formula works for other functions, for example
- $E (3 X^{2} + 5) = 3 E (X^{2}) + 5$
  - Note that $E (X^{2})$ does not simplify to ${(E (X))}^{2}$
- Simplify algebra before taking expectations
  - $E ({(X - 1)}^{2}) = E (X^{2} - 2 X + 1) = E (X^{2}) - 2 E (X) + 1$
The variance formula also works for other functions, for example
- $Var (4 X^{3}) = 16 Var (X^{3})$
  - Where $Var (X^{3})$ means $E (X^{6}) - {(E (X^{3}))}^{2}$
- Simplify algebra before taking variances
  - $Var ({(1 - 3 X)}^{2}) = Var (1 - 6 X + 9 X^{2}) = 36 Var (X) + 81 Var (X^{2})$

$X$ is a random variable such that $E (X) = 5$ and $Var (X) = 4$ .

Find the value of:

(i)

E (3 X + 5)

(ii)

Var (3 X + 5)

(iii)

Var (2 - X)

ax-and-b-we