Does adding a constant b change the variance?

No; a shift does not change the spread.

Standard deviation of aX+b?

|a|\times the standard deviation of X.

Why can a negative a not give a negative variance?

The variance is multiplied by a^2, which is positive.

What happens to probabilities when Y=aX+b?

They are unchanged; only the values change.

How do you standardise a variable with mean \mu and standard deviation \sigma?

Use Y=\frac{X-\mu}{\sigma}, i.e. a=\frac1\sigma, b=-\frac\mu\sigma.

How can you check results for Y=aX+b directly?

List the values y=ax+b with the same probabilities, then find E(Y) and E(Y^2).

Linear functions of a DRVAQA A-Level Further Maths: Flashcards

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Formula for $E(aX+b)$?

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Formula for $E(aX+b)$ ?: $aE(X)+b$ .
Formula for $\mathrm{Var}(aX+b)$ ?: $a^2\mathrm{Var}(X)$ .
Does adding a constant $b$ change the variance?: No; a shift does not change the spread.
Standard deviation of $aX+b$ ?: $|a|\times$ the standard deviation of $X$ .
Why can a negative $a$ not give a negative variance?: The variance is multiplied by $a^2$ , which is positive.
Why does $E(aX+b)=aE(X)+b$ use $\sum p_i=1$ ?: $\sum bp_i=b\sum p_i=b$ .
What happens to probabilities when $Y=aX+b$ ?: They are unchanged; only the values change.
$E(X)=5$ . Find $E(3X+2)$ .: $3(5)+2=17$ .
$\mathrm{Var}(X)=4$ . Find $\mathrm{Var}(3X+2)$ .: $9\times4=36$ .
$\mathrm{Var}(X)=4$ . Find the standard deviation of $2-4X$ .: $4\times2=8$ .
How do you standardise a variable with mean $\mu$ and standard deviation $\sigma$ ?: Use $Y=\frac{X-\mu}{\sigma}$ , i.e. $a=\frac1\sigma$ , $b=-\frac\mu\sigma$ .
How can you check results for $Y=aX+b$ directly?: List the values $y=ax+b$ with the same probabilities, then find $E(Y)$ and $E(Y^2)$ .

Exam questions on Linear functions of a DRV

The discrete random variable $X$ has $E(X)=5$ and $\mathrm{Var}(X)=4$ . The random variable $Y$ is defined by $Y=3X+2$ .
Find the standard deviation of $2-4X$ .2 marks
The number of special meals, $X$ , sold at a café in one day has $E(X)=8.5$ and $\mathrm{Var}(X)=2.25$ . The daily profit, in pounds, is $T=6X-20$ .
The café changes its pricing so that the daily profit becomes $T=8X-30$ . Find $E(T)$ and $\mathrm{Var}(T)$ .2 marks
The noon temperature $C$ ( $^\circ$ C) in a greenhouse is modelled as a discrete random variable with $E(C)=22$ and standard deviation $3$ . The temperature in degrees Fahrenheit is $F=1.8C+32$ .
Find $E(F)$ and $\mathrm{Var}(F)$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).