HL · Further probability 1 · Transforming one random variable: E(aX + b) and Var(aX + b)

Module 31 of 47Lesson 1 of 9 · HL · Further probability 1 · Transforming one random variable: E(aX + b) and Var(aX + b)

Further probability: transformed variables, X̄, Poisson & Markov chains · Essential understanding 1

Transforming one random variable: E(aX + b) and Var(aX + b)

IB AI AHL 4.14with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: If you multiply a random variable by

a

and add

b

, the mean does exactly the same thing, but the variance is only affected by the multiplying, and it is multiplied by

a^2

. This is how we change units, add fixed charges and rescale scores.

By the end of this lesson you can:

use $E(aX + b) = aE(X) + b$ to find a new mean
use $\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)$ to find a new variance and standard deviation
explain why adding a constant does not change the spread, and why a negative $a$ never gives a negative variance
find $E(X)$ and $\mathrm{Var}(X)$ from a probability table with the GDC, then transform them
work backwards to find $a$ and $b$ from a given mean and variance

🧑🏾 Imagine this…

A taxi driver in Francistown charges P20 as a starting fee plus P3 for every kilometre. The number of kilometres

X

changes from trip to trip. So the money earned on a trip is

Y = 3X + 20

. If the average trip is 10 km, the average fare is

3 \times 10 + 20 = \text{P}50

. Easy: the mean follows the same rule as the fare. But how spread out are the fares? The starting fee P20 is the same on every trip, so it moves every fare up together: no extra spread. The P3 per km stretches the differences between trips by 3. That is the whole lesson.

E(X)

the expected value (long-run mean) of

X

\mathrm{Var}(X)

the variance: the mean squared distance of

X

from

E(X)

Standard deviation

\sigma = \sqrt{\mathrm{Var}(X)}

. It is in the same units as

X

; variance is in squared units.

Y = aX + b

a linear transformation of

X

: multiply every value by

a

(scale), then add

b

(shift).

★ The two rules (in your IB formula booklet)

$E(aX + b) = aE(X) + b$
$\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)$ , so the standard deviation of $aX + b$ is $|a|$ times the standard deviation of $X$ .
Shifting (adding $b$ ) moves the mean by $b$ but leaves the variance and standard deviation alone.
Scaling (multiplying by $a$ ) multiplies the mean by $a$ , the standard deviation by $|a|$ and the variance by $a^2$ .
From a probability table: $E(X) = \sum x\,P(X = x)$ and $\mathrm{Var}(X) = E(X^2) - [E(X)]^2$ .
Special cases: $E(b) = b$ , $\mathrm{Var}(b) = 0$ (a constant never varies), $E(aX) = aE(X)$ .

⚠️ Mistakes that lose marks

Do not add $b$ to the variance, and do not forget to square $a$ . $\mathrm{Var}(3X + 2)$ is $9\,\mathrm{Var}(X)$ , not $3\,\mathrm{Var}(X) + 2$ .
A negative $a$ flips the mean but never makes the variance negative: $\mathrm{Var}(5 - 2X) = (-2)^2\,\mathrm{Var}(X) = 4\,\mathrm{Var}(X)$ . A variance can never be below 0.
Keep variance and standard deviation apart. If the question gives the standard deviation, square it before using the variance rule.
When you change units, change both the mean and the spread, and give the new units in your answer.

💡 On the TI-Nspire CX II

To get

E(X)

and

\mathrm{Var}(X)

from a probability table, put the values in list x and the probabilities in list p. Run menu💻 click menu on keypad▸6: Statistics▸1: Stat Calculations▸1: One-Variable Statistics with X1 List = x and Frequency List = p. Then

E(X)

is x̄, and

\mathrm{Var}(X)

is σx² (use the population

\sigma x

, not

sx

), because the probabilities add to 1.

Prof. Kitso says

Say it in words: “adding changes the centre, multiplying changes the spread.” Variance is a squared quantity, so multiplying by 3 makes it 9 times bigger. If you can say that sentence, you will not mix up the two rules in an exam.

🧪 Try it yourselfTry it: shift and stretch a data set

These are eight daily maximum temperatures (°C). Use the transform controls to add k: the mean moves, but σx does not. Then multiply by k: the mean and σx both multiply, and the variance multiplies by k². This is the same behaviour as for a random variable.

Worked examples

Example 1.1Mean and variance of a transformed variable

At a taxi rank in Francistown, the number of passengers

X

on a trip has

E(X) = 8

and

\mathrm{Var}(X) = 2.5

. A driver earns

Y = 4X + 3

pula on a trip (P4 for each passenger and a P3 fixed fee).

(a) Find $E(Y)$ .
(b) Find $\mathrm{Var}(Y)$ and the standard deviation of $Y$ .

🧠 Think it through

Here $a = 4$ and $b = 3$ .
Mean: do to $E(X)$ exactly what the formula does to $X$ .
Variance: the fixed fee $b = 3$ disappears. Square the multiplier: $4^2 = 16$ .
Standard deviation: square root of the variance (or $4 \times \sqrt{2.5}$ ).

✍️ By hand

(a)

E(Y) = 4\,E(X) + 3 = 4(8) + 3 = 35

(b)

\mathrm{Var}(Y) = 4^2\,\mathrm{Var}(X) = 16 \times 2.5 = 40

\sigma_Y = \sqrt{40} = 6.32

🔢 On your TI-Nspire CX II

Check the standard deviation on a Calculator page: type sqrt(40) and 4*sqrt(2.5) — they must agree.
📍 Where to find it on the TI-Nspire
⌨️ square root: ctrl then x² on the keypad, or type sqrt(

✅ Answer (a)

E(Y) = 35

, so P35 per trip on average. (b)

\mathrm{Var}(Y) = 40

(pula²), standard deviation = P6.32.

Write the units: variance is in pula squared, but the standard deviation is in pula.

Example 1.2A negative multiplier

A generator has a 50-litre tank. It uses 2 litres of fuel for every hour it runs. The running time

X

hours in a week has

E(X) = 20

and

\mathrm{Var}(X) = 3.5

. The fuel left at the end of the week is

Y = 50 - 2X

litres.

(a) Find the expected fuel left.
(b) Find $\mathrm{Var}(Y)$ and the standard deviation of $Y$ .
(c) Explain why $\mathrm{Var}(Y)$ is positive even though the multiplier is $-2$ .

🧠 Think it through

Write $Y = (-2)X + 50$ , so $a = -2$ and $b = 50$ .
The mean follows the formula, negative sign and all.
For the variance, square $a$ : $(-2)^2 = 4$ .

✍️ By hand

(a)

E(Y) = -2(20) + 50 = 10

(b)

\mathrm{Var}(Y) = (-2)^2 \times 3.5 = 14

\sigma_Y = \sqrt{14} = 3.74

🔢 On your TI-Nspire CX II

Type (-2)^2*3.5 and sqrt(14) on a Calculator page.
📍 Where to find it on the TI-Nspire
⌨️ square root: ctrl then x² on the keypad, or type sqrt(

✅ Answer (a) 10 litres. (b)

\mathrm{Var}(Y) = 14

litres², standard deviation 3.74 litres. (c) Squaring

-2

gives 4. Variance measures squared distances from the mean, and these can never be negative. Running 1 hour more or 1 hour less than average changes the fuel left by 2 litres either way, so the spread is positive.

A common wrong answer is

\mathrm{Var}(Y) = -2 \times 3.5 = -7

. A negative variance is impossible, so always check your sign.

Example 1.3Changing units

The daily maximum temperature

C

in Gaborone in January has mean 28 °C and standard deviation 3.2 °C. A tourist website shows temperatures in °F, using

F = 1.8C + 32

(a) Find the mean temperature in °F.
(b) Find the standard deviation and the variance in °F.
(c) Does adding 32 make the temperatures more spread out? Explain.

🧠 Think it through

$a = 1.8$ , $b = 32$ .
Standard deviation is multiplied by $|a|$ only. Variance is multiplied by $a^2$ .
Check: the variance should equal (new standard deviation)².

✍️ By hand

(a)

E(F) = 1.8(28) + 32 = 82.4

(b)

\sigma_F = 1.8 \times 3.2 = 5.76

\mathrm{Var}(F) = 1.8^2 \times 3.2^2 = 33.2

🔢 On your TI-Nspire CX II

Type 1.8*28+32, 1.8*3.2 and (1.8*3.2)^2.

✅ Answer (a) 82.4 °F. (b) Standard deviation 5.76 °F; variance 33.2 °F². (c) No. Adding 32 moves every temperature up by the same amount, so the differences between days do not change. The spread grows only because of the 1.8.

Notice the new variance is NOT

1.8 \times 3.2^2 + 32

. Variance uses

a^2

and ignores

b

Example 1.4From a probability table

A mechanic in Maun repairs

X

cars on a working day, with this probability distribution.

x	1	2	3	4	5
P(X = x)	0.10	0.25	0.35	0.20	0.10

He earns

Y = 300X - 100

pula per day (P300 per car, minus P100 for the workshop).

(a) Find $E(X)$ and $\mathrm{Var}(X)$ .
(b) Hence find $E(Y)$ and $\mathrm{Var}(Y)$ .
(c) Find the standard deviation of his daily earnings.

🧠 Think it through

Use list x and list p, then One-Variable Statistics with the probabilities as the frequency list. Because the probabilities add to 1, x̄ is $E(X)$ and $\sigma x^2$ is $\mathrm{Var}(X)$ .
Apply the two rules with $a = 300$ , $b = -100$ .

✍️ By hand
(a) 
(b) 

🔢 On your TI-Nspire CX II

On a Lists & Spreadsheet page enter the values in column x and the probabilities in column p.
Run menu💻 click menu on keypad▸6: Statistics▸1: Stat Calculations▸1: One-Variable Statistics with X1 List = x and Frequency List = p. Read x̄ and σx.
x̄ = E(X); σx = 1.11692, so Var(X) = σx² = 1.2475
Now transform: type 300*2.95-100, 300^2*1.2475 and 300*1.11692.

✅ Answer (a)

E(X) = 2.95

\mathrm{Var}(X) = 1.2475

. (b)

E(Y) = \text{P}785

\mathrm{Var}(Y) = 112\,275

. (c) Standard deviation

= \sqrt{112\,275} = \text{P}335

On the real calculator the line “sx” may display undef because n = 1 (the probabilities add to 1). Ignore it and use σx.

Example 1.5Finding a and b

The random variable

X

has

E(X) = 6

and

\mathrm{Var}(X) = 4

. The random variable

Y = aX + b

has

E(Y) = 20

and

\mathrm{Var}(Y) = 36

(a) Find the two possible values of $a$ .
(b) Given that $a > 0$ , find $b$ .
(c) Given instead that $a < 0$ , find $b$ .

🧠 Think it through

Variance gives an equation in $a^2$ : $a^2 \times 4 = 36$ .
Then use the mean equation $6a + b = 20$ once you know $a$ .
Do both signs: a square root has a positive and a negative answer.

✍️ By hand

(a)

a^2\,\mathrm{Var}(X) = 36 \Rightarrow 4a^2 = 36 \Rightarrow a^2 = 9

a = 3

a = -3

(b)

6(3) + b = 20 \Rightarrow b = 2

(c)

6(-3) + b = 20 \Rightarrow b = 38

🔢 On your TI-Nspire CX II

Check both: type 3*6+2 and (-3)*6+38. Both must give 20.

✅ Answer (a)

a = 3

a = -3

. (b)

b = 2

. (c)

b = 38

Exam questions often say “

a > 0

” to force a single answer. If there is no such condition, give both.

Challenge questions

⭐ Challenge 1.6Working backwardsPaper 1 style

The random variable

X

is such that

E(4X - 3) = 17

and

\mathrm{Var}(4X - 3) = 28.8

(a) Find $E(X)$ .
(b) Find $\mathrm{Var}(X)$ .
(c) Find $E(2 - 5X)$ and $\mathrm{Var}(2 - 5X)$ .

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Put the rule $E(4X - 3) = 4E(X) - 3$ equal to 17.
The variance rule gives $16\,\mathrm{Var}(X) = 28.8$ .
For $2 - 5X$ : $a = -5$ , $b = 2$ .

✍️ By hand

(a)

4E(X) - 3 = 17 \Rightarrow E(X) = 5

(b)

16\,\mathrm{Var}(X) = 28.8 \Rightarrow \mathrm{Var}(X) = 1.8

(c)

E(2 - 5X) = 2 - 25 = -23

\mathrm{Var}(2 - 5X) = 25(1.8) = 45

🔢 On your TI-Nspire CX II

Check with a Calculator page: (17+3)/4 and 28.8/16 and 25*1.8.

✅ Answer (a) 5. (b) 1.8. (c)

E(2 - 5X) = -23

\mathrm{Var}(2 - 5X) = 45

⭐ Challenge 1.7Bus times and standardisingPaper 2 style

The journey time

T

minutes of a bus from Kasane to the border has mean 42 and standard deviation 6.

(a) Let $H = \dfrac{T}{60}$ be the time in hours. Find the mean and standard deviation of $H$ .
(b) The fare for a charter trip is $C = 12 + 0.75T$ pula. Find $E(C)$ , $\mathrm{Var}(C)$ and the standard deviation of $C$ .
(c) Find constants $a$ and $b$ such that $Z = aT + b$ has mean 0 and variance 1.
(d) Explain in one sentence what $Z = 0$ and $Z = 1$ mean for a journey.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

(a) $H = \dfrac{1}{60}T + 0$ , so $a = \dfrac{1}{60}$ .
(b) $a = 0.75$ , $b = 12$ . Square the standard deviation first if you need the variance.
(c) Two conditions: $42a + b = 0$ and $36a^2 = 1$ . Take $a > 0$ .
(d) Think about distance from the mean in standard deviations.

✍️ By hand
(a)  h,  h
(b) 
(c) , and 
, so 

🔢 On your TI-Nspire CX II

Check on a Calculator page: 12+0.75*42, 0.75^2*36 and 6/60.

✅ Answer (a) Mean 0.7 h, standard deviation 0.1 h. (b)

E(C) = \text{P}43.50

\mathrm{Var}(C) = 20.25

, standard deviation P4.50. (c)

a = \dfrac{1}{6}

b = -7

, i.e.

Z = \dfrac{T - 42}{6}

. (d)

Z = 0

is an average journey (42 min);

Z = 1

is a journey exactly one standard deviation (6 min) longer than average, i.e. 48 min.

Next: Lesson 2 · Combining independent random variables

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL HL · Further probability 1 · Transforming one random variable: E(aX + b) and Var(aX + b)

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

Transforming one random variable: E(aX + b) and Var(aX + b)

Worked examples

Challenge questions