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H2 Maths Notes 6 2 Discrete Random Variables

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H2 Maths Notes (JC 1-2): 6.2) Discrete Random Variables

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07 Oct 2025, 00:00 Z

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Q: What does H2 Maths Notes (JC 1-2): 6.2) Discrete Random Variables cover?
A: Expectation, variance, binomial modelling, and calculator workflows for H2 discrete distributions.

Study cadence
Re-derive the expectation and variance formulas from first principles once per week so you remember why each term appears. Keep a small table template in your notes for probability mass functions (PMFs) so you can slot in values quickly during exams.

Core Concepts

A discrete random variable $X$ takes countable values $x_1, x_2, \dots$ with probabilities $P\left(X = x_{\mathrm{i}}\right)$

Part 25 of 32

Previous topicH2 Maths Notes (JC 1-2): 6.1) Probability Next topicH2 Maths Notes (JC 1-2): 6.3) Normal Distribution

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Expectation:

E(X) = \sum x_i P\left( X = x_{\mathrm{i}}\right)

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Second moment:

E(X^2) = \sum x_i^2 P\left( X = x_{\mathrm{i}}\right)

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Variance:

\operatorname{Var}(X) = E(X^2) - [E(X)]^2

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Standard deviation: square root of

\operatorname{Var}(X)

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Binomial Model

Use $X \sim \operatorname{Bin}(n, p)$ when you have $n$ independent Bernoulli trials with success probability $p$ .

Probability mass: $P\left( X = r \right) = \binom{n}{r} p^r (1 - p)^{n - r}$ .
Expectation: $E(X) = n p$ .
Variance: $\operatorname{Var}(X) = n p (1 - p)$ .
Conditions: fixed $n$ , constant $p$ , independent trials, success/failure outcomes.

Example -- Hiring round

A firm interviews $n = 12$ candidates. Each has a 0.3 probability of accepting an offer. Let $X$ be the number who accept.

$P\left( X = 4 \right) = \binom{12}{4} 0.3^4 0.7^8 \approx 0.231$ .
$P\left( X \geq 2 \right) = 1 - \bigl[P\left( X = 0 \right) + P\left( X = 1 \right)\bigr] \approx 0.896$
$E(X) = 12 \times 0.3 = 3.6$ .
$\operatorname{Var}(X) = 12 \times 0.3 \times 0.7 = 2.52$ .

Custom PMFs

When the PMF is tabulated (common in H2 questions):

$x$	$0$	$1$	$2$	$3$
$P\left( X = x \right)$	$0.1$	$0.3$	$0.4$	$k$

Use the total probability condition to find $k = 0.2$ .
$E(X) = 0 \times 0.1 + 1 \times 0.3 + 2 \times 0.4 + 3 \times 0.2 = 1.7$ .
$E(X^2) = 0^2 \times 0.1 + 1^2 \times 0.3 + 4 \times 0.4 + 9 \times 0.2 = 3.5$
$\operatorname{Var}(X) = 3.5 - 1.7^2 = 0.61$ .

Transformations

For $Y = aX + b$ : $E(Y) = a E(X) + b$ and $\operatorname{Var}(Y) = a^2 \operatorname{Var}(X)$ .
Linear combinations: for independent $X$ and $Y$ , $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)$

Example -- Bonus payouts

Let $X$ be binomial $\operatorname{Bin}(8, 0.4)$ . Each success triggers a 200 dollar bonus plus a fixed 150 dollar stipend. Total payout $T = 200 X + 150$ .

$E(T) = 200 \times 8 \times 0.4 + 150 = 790$ dollars.
$\operatorname{Var}(T) = 200^2 \times 8 \times 0.4 \times 0.6 = 76800$ dollars squared.
Standard deviation $\approx 277$ dollars.

Calculator Workflows

Casio: BPD (binomial probability distribution) for individual terms, BCD (binomial cumulative distribution) for cumulative sums. Store n and p in variables for quick reuse.
TI: binompdf(n, p, r) and binomcdf(n, p, r) functions in the DISTR menu. For PMF tables, program a short list-based routine to multiply L1 (values) and L2 (probabilities).
Cross-check expectation and variance using calculator built-ins but still show the manual summation in working.

Exam Watch Points

State whether a binomial model is appropriate; include independence and constant probability statements.
When a question mentions "at least one" or "no more than", use complements to reduce calculator input.
For custom PMFs, ensure probabilities sum to 1 before computing expectation.
Payout or cost problems often require a linear transformation-write the transformation before substituting.

Practice Quiz

Check your understanding of discrete PMFs, expectation/variance, and binomial modelling decisions.

Quick Revision Checklist

Identify when binomial modelling fits and articulate the assumptions.
Compute expectation and variance efficiently, including from tables.
Execute linear transformations and interpret the resulting mean and variance.
Use calculator functions accurately while documenting the method in words.
Spot when normal approximations are appropriate (preview for Topic 6.3).

Next steps: follow the H2 Maths notes hub into Topic 6.3 - Normal distribution for approximation workflows.

H2 Maths Notes (JC 1-2): 6.2) Discrete Random Variables

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