AHL 4.14 β€” Continuous & Discrete Random Variables, Variance & Linear Transformations

Topic Preview Short Description
Discrete Random Variables Variables that take a finite or countably infinite set of values, each with an associated probability.
Continuous Random Variables Variables that take infinitely many values over an interval, with probabilities defined by areas.
Variance & Standard Deviation Measures describing how spread out values are around the mean.
Linear Transformations Rules governing how mean and variance change when variables are scaled or shifted.

πŸ“Œ 1. Variance of a Discrete Random Variable

Definition: For a discrete random variable X taking values x₁, xβ‚‚, … with probabilities P(X = x):

E(X) = Ξ£[x Β· P(X = x)]
Var(X) = E(XΒ²) βˆ’ [E(X)]Β²

Key ideas

  • Variance measures the expected squared deviation of values from the mean.
  • It quantifies spread, not central tendency.
  • Squaring ensures deviations above and below the mean contribute positively.
  • A larger variance indicates greater unpredictability.
  • Variance is sensitive to extreme values.

Worked example

  • The probabilities must sum to 1 before calculations begin.
  • E(X) is calculated first using weighted averages.
  • E(XΒ²) must be computed separately using squared values.
  • Only after summing do we subtract [E(X)]Β².
  • This two-step structure is essential for full marks.

🧠 Examiner Tip

  • Never square probabilities β€” only square the values of X.
  • Always calculate E(XΒ²) separately.
  • Writing Var(X) = Ξ£(x βˆ’ ΞΌ)Β²P(x) is allowed but slower.
  • Most lost marks come from skipping E(XΒ²).
  • State final answers clearly with correct units.

πŸ“Œ 2. Continuous Random Variables & Probability Density Functions

Core principles

  • Continuous variables take infinitely many possible values.
  • Probabilities are defined by areas under a curve, not by point values.
  • The probability density function f(x) must satisfy f(x) β‰₯ 0.
  • The total area under f(x) over all x equals 1.
  • P(X = exact value) = 0 for all continuous variables.

P(a ≀ X ≀ b) = ∫ab f(x) dx

🌍 Real-World Connection

  • Reaction times are measured on a continuous scale.
  • Heights and weights are modelled using continuous distributions.
  • Manufacturing tolerances rely on continuous models.
  • Medical measurements often assume continuity.
  • Continuous models allow smooth probability estimates.

πŸ“Œ 3. Mode & Median of Continuous Random Variables

Key interpretations

  • The mode is the x-value where the pdf reaches its maximum height.
  • The median divides the total probability into two equal halves.
  • The median satisfies βˆ«β‚‹βˆžα΅ f(x) dx = 1/2.
  • Mode and median need not coincide.
  • Skewed distributions separate mean, median, and mode.

πŸ“Œ 4. Mean, Variance & Standard Deviation

  • Variance is always non-negative.
  • Standard deviation is the square root of variance.
  • Standard deviation is measured in original units.
  • Variance simplifies algebraic manipulation.
  • Fair games satisfy E(X) = 0.

πŸ“± GDC Tips

TI-Nspire CX II

  • Enter values in Lists & Spreadsheet.
  • Store probabilities in a second column.
  • Use Menu β†’ Statistics β†’ Stat Calculations β†’ One-Variable Statistics.
  • Verify probabilities sum to 1 before calculating.

Casio fx-CG50 / CG100

  • Enter x-values in List1 and probabilities in List2.
  • Use STAT β†’ CALC β†’ 1-VAR.
  • Ensure frequency column is set correctly.
  • Quote calculator output clearly in exams.

πŸ“Œ 5. Effect of Linear Transformations

E(aX + b) = aE(X) + b
Var(aX + b) = aΒ²Var(X)

  • Multiplying by a scales both mean and spread.
  • Adding b shifts the distribution horizontally.
  • Variance is unaffected by addition.
  • Standard deviation scales by |a|.
  • These rules avoid unnecessary algebra.

πŸ“ Paper 2 Strategy

  • Apply transformation rules directly.
  • Do not expand random variables unnecessarily.
  • State final mean and variance clearly.
  • Use correct notation throughout.
  • This saves time and reduces error risk.

πŸ” TOK Perspective

Are probability models discoveries about the world, or tools we invent to manage uncertainty?
How does mathematical abstraction shape what we consider β€œrandom”?

πŸ“Œ Practice Questions β€” Continuous & Discrete Random Variables

Multiple Choice Questions

MCQ 1
Which of the following statements correctly distinguishes a discrete random variable from a continuous random variable?

  • A. A discrete random variable has probabilities defined using areas under a curve
  • B. A continuous random variable can only take integer values
  • C. A discrete random variable takes countable values with assigned probabilities
  • D. A continuous random variable has P(X = x) > 0 for some x
Answer & Explanation

Correct answer: C

A discrete random variable takes a countable set of values, and each value has an explicitly defined probability.
Continuous random variables instead use probability density functions and probabilities over intervals, not at individual points.

MCQ 2
A random variable X has Var(X) = 5. What is the variance of the random variable Y = βˆ’4X + 9?

  • A. 5
  • B. 20
  • C. 80
  • D. 144
Answer & Explanation

Correct answer: C

For a linear transformation Y = aX + b, variance transforms as Var(Y) = aΒ²Var(X).
Here a = βˆ’4, so Var(Y) = (βˆ’4)Β² Γ— 5 = 16 Γ— 5 = 80.
The constant +9 does not affect the variance.

MCQ 3
For a continuous random variable X with probability density function f(x), which condition must always be satisfied?

  • A. f(x) ≀ 1 for all x
  • B. βˆ«β‚‹βˆžβΊβˆž f(x) dx = 1
  • C. P(X = a) > 0 for some a
  • D. The mean must equal the median
Answer & Explanation

Correct answer: B

For any valid probability density function, the total area under the curve across all real values must equal 1.
This represents total probability.
The other statements are not required for all continuous distributions.

Short Answer Questions

Short Question 1
Explain why adding a constant to a random variable changes the mean but not the variance.

Model Answer

Adding a constant shifts every value of the random variable and the mean by the same amount.
However, variance measures spread around the mean, and these distances remain unchanged.
Therefore, variance is unaffected by addition.

Short Question 2
State one reason why probabilities for continuous random variables are calculated over intervals rather than exact values.

Model Answer

A continuous random variable can take infinitely many values within any interval, so an individual value has zero probability.
Only intervals have non-zero probability, calculated using areas under the density function.

Long Answer Questions (IB AIHL Style)

Long Question 1

A discrete random variable X has the following probability distribution:

X: 0  1  2  3
P(X): 0.1 0.3 0.4 0.2

(a) Find the value of E(X).
(b) Find the value of Var(X).
(c) Hence find the standard deviation of X.
(d) Interpret the standard deviation in the context of the distribution.

Full Worked Solution

(a) Mean

E(X) = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 1.7

(b) Variance

E(XΒ²) = 0Β²(0.1) + 1Β²(0.3) + 2Β²(0.4) + 3Β²(0.2) = 3.7

Var(X) = 3.7 βˆ’ (1.7)Β² = 0.81

(c) Standard deviation

SD(X) = √0.81 = 0.9

(d) Interpretation

The standard deviation indicates that values of X typically differ from the mean by about 0.9 units,
showing a moderate spread around 1.7.

Long Question 2

Let X be a random variable with E(X) = 6 and Var(X) = 2.
Define Y = 5 βˆ’ 3X.

(a) Find E(Y).
(b) Find Var(Y).
(c) Describe the effect of this transformation on the distribution of X.
(d) Explain why the sign of the multiplier does not affect the variance.

Full Worked Solution

(a) E(Y) = 5 βˆ’ 3(6) = βˆ’13

(b) Var(Y) = (βˆ’3)Β² Γ— 2 = 18

(c)
The negative multiplier reflects the distribution, the factor 3 stretches it,
and the constant 5 shifts it without changing spread.

(d)
Variance depends on squared deviations, so the sign disappears when squaring the multiplier.