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CCEA GCE Mathematics (2210) · Statistical distributions
Mini-Lesson

Statistical distributions

This mini-lesson covers discrete random variables and E(X), the binomial distribution B(n, p) — when it applies, its probabilities, mean and variance — and the normal distribution, including standardising with z = (x − μ)/σ.

Where this sits: Unit AS 2: Applied Mathematics — statistical distributions: discrete random variables and the binomial distribution as a model. Unit A2 2 — the normal distribution and its use as a model.

Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.

Discrete random variables

Probability distributions and E(X)

A discrete random variable X takes separate values, each with a probability. The probabilities must sum to 1.

E(X) = Σ x P(X = x)the "expected value" = the long-run mean — it need not be a possible value of X
Worked example

X takes 1, 2, 3 with probabilities 0.1, 0.4, 0.5 (check: they sum to 1 ✓).

E(X) = 1(0.1) + 2(0.4) + 3(0.5) = 0.1 + 0.8 + 1.5 = 2.4

Calculate

Expected value

1X takes the values 1, 2 and 3 with probabilities 0.1, 0.4 and 0.5. Find E(X).
Hint: E(X) = Σ x P(X = x) = 1(0.1) + 2(0.4) + 3(0.5) = 0.1 + 0.8 + 1.5.
Binomial

The binomial distribution B(n, p)

X ~ B(n, p) counts successes when all four conditions hold:

  • a fixed number n of trials;
  • each trial has only two outcomes (success/failure);
  • the trials are independent;
  • p, the probability of success, is constant.
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ · E(X) = np · Var(X) = np(1 − p)your calculator gives binomial pdf and cdf — but you must know these formulae
Worked example — X ~ B(10, 0.3)

P(X = 3) = ¹⁰C₃ (0.3)³ (0.7)⁷ = 120 × 0.027 × 0.08235 = 0.267 (3 d.p.)

E(X) = 10 × 0.3 = 3 · Var(X) = 10 × 0.3 × 0.7 = 2.1

Calculate

Binomial probability

2X ~ B(10, 0.3). Find P(X = 3) to 3 decimal places.
Hint: P(X = 3) = ¹⁰C₃ × 0.3³ × 0.7⁷ = 120 × 0.027 × 0.0823543 = 0.2668.
Calculate

Binomial variance

3For X ~ B(10, 0.3), find the variance.
Hint: Var(X) = np(1 − p) = 10 × 0.3 × 0.7. (The mean is np = 3.)
Quick check

Is it binomial?

?Which situation is NOT modelled by a binomial distribution?
Normal

The normal distribution

X ~ N(μ, σ²) is the classic bell curve: symmetric about the mean μ, with spread set by σ.

  • mean = median = mode = μ;
  • about 68% of data lie within 1σ of μ, 95% within 2σ, 99.7% within 3σ;
  • points of inflection at μ ± σ.
z = (x − μ) / σstandardising: converts any normal value into the standard normal Z ~ N(0, 1)
Worked example — X ~ N(50, 8²)

For x = 62: z = (62 − 50)/8 = 12/8 = 1.5

P(X < 62) = P(Z < 1.5) = 0.9332

So P(X > 62) = 1 − 0.9332 = 0.0668.

Calculate

Standardise

4X ~ N(50, 8²). Find the z-value for x = 62.
Hint: z = (x − μ)/σ = (62 − 50) ÷ 8 = 12 ÷ 8. Note σ = 8, not 64 — the 8² in the notation is the VARIANCE.
Calculate

Normal probability

5X ~ N(50, 8²). Find P(X < 62) to 4 decimal places.
Hint: Standardise: z = 1.5. Then P(Z < 1.5) = 0.9332 from the normal tables or your calculator.
Sort it

Which model?

Tap a situation, then tap the distribution that models it.

🎯 Binomial

🔔 Normal

🚫 Neither

Quick check

Variance notation trap

?X ~ N(100, 25). What is the standard deviation?
Match it

Match the distribution fact

Tap the item on the left, then its formula on the right.

Statement
Answer
Quick check

Symmetry

?X ~ N(μ, σ²). What is P(X > μ)?
Quick check

Binomial conditions

?A biased coin with P(head) = 0.6 is tossed 15 times, and X is the number of heads. Which statement is true?
Examiner traps

Distribution pitfalls

  • N(μ, σ²) gives the VARIANCE. N(100, 25) has σ = 5, not 25.
  • Binomial needs a fixed n. "Trials until the first success" is not binomial.
  • P(X ≤ r) vs P(X < r). For a discrete X these differ: P(X < 5) = P(X ≤ 4).
  • Var(X) = np(1 − p), not np.
  • The normal is continuous, so P(X = a) = 0 and P(X < a) = P(X ≤ a).
Recap

The big ideas to know

Discrete RV: probabilities sum to 1 · E(X) = Σ x P(X = x)

Binomial B(n, p): fixed n, two outcomes, independent trials, constant p

P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ · mean = np · variance = np(1 − p)

Normal N(μ, σ²): symmetric bell curve; 68% / 95% / 99.7% within 1, 2, 3 sd

Standardise: z = (x − μ)/σ, then use Z ~ N(0, 1) — and remember σ² is the VARIANCE

These two distributions supply the models tested in the next topic: hypothesis testing. Press Finish to see your score.

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