Statistical distributions (A2 Unit 4: Applied Mathematics B) — WJEC A-Level Mathematics
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Statistical distributions (A2 Unit 4: Applied Mathematics B) explained
Continuous random variables are modelled by probability density functions f(x).
Read the full explanation
The continuous uniform (rectangular) distribution X ~ U(a, b) models a variable equally likely to fall anywhere within interval [a, b], with density f(x) = 1/(b − a) for a ≤ x ≤ b and 0 elsewhere; its mean is (a + b)/2 and variance is (b − a)²/12. The Normal distribution X ~ N(μ, σ²) models continuous variables exhibiting a symmetrical, bell-shaped distribution about mean μ with variance σ². Total area under both density curves equals 1, and the probability of any single point is zero: P(X = c) = 0.
Your focus
- Define and sketch the probability density function for a continuous uniform distribution X ~ U(a, b).
- Calculate interval probabilities for uniform distributions using geometric rectangular areas.
- Describe the fundamental characteristics of the Normal distribution: bell shape, symmetry, and parameters μ and σ².
Statistical distributions (A2 Unit 4: Applied Mathematics B) exam tips
Marking Points
- defining or sketching the probability density function of X ~ U(a, b) with height 1 / (b − a)
- calculating probabilities for a uniform distribution as rectangular areas: P(c ≤ X ≤ d) = (d − c) / (b − a)
- stating the properties of the Normal distribution: bell-shaped, symmetrical about mean μ, total area 1
Examiner Tips
- 💡For uniform distributions, sketch the rectangle: area = base × height gives the probability directly.
- 💡Remember that for continuous distributions, P(X ≤ c) = P(X < c) because the boundary point has zero area.
Common Mistakes
- forgetting that f(x) = 1/(b − a) is a probability density, not a discrete point probability
- calculating P(X = c) for a continuous variable instead of recognising that point probabilities are zero