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A-Level Further Maths Flashcards: Further Statistics

A-Level Further Maths — Further Statistics Flashcards

20 flashcards with spaced repetition. Press Space to flip, then rate your recall (1—4).

What These Flashcards Cover

These flashcards extend A-Level statistics into more formal and advanced territory. You will work with probability distributions more rigorously, learn about continuous random variables, and apply hypothesis testing with the tt-distribution and χ2\chi^{2} tests.

Key areas:

  • Probability Mass Functions (PMF): For a discrete random variable XX, the PMF gives P(X=x)P(X = x). Properties: ∑P(X=x)=1\sum P(X = x) = 1 and all probabilities are non-negative. Expected value E(X)=∑x⋅P(X=x)E(X) = \sum x \cdot P(X = x). Variance Var(X)=E(X2)−[E(X)]2\text{Var}(X) = E(X^{2}) - [E(X)]^{2}.
  • Probability Density Functions (PDF): For a continuous random variable, f(x)f(x) is the density, not a probability. P(a≤X≤b)=∫abf(x) dxP(a \leq X \leq b) = \int_{a}^{b} f(x)\,dx. Properties: f(x)≥0f(x) \geq 0 and ∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x)\,dx = 1. E(X)=∫xf(x) dxE(X) = \int x f(x)\,dx.
  • Coding and Linear Transformations: If Y=aX+bY = aX + b, then E(Y)=aE(X)+bE(Y) = aE(X) + b and Var(Y)=a2Var(X)\text{Var}(Y) = a^{2}\text{Var}(X). Standardising: Z=X−μσZ = \frac{X - \mu}{\sigma} gives Z∼N(0,1)Z \sim N(0,1).
  • Continuous Distributions: The uniform distribution U(a,b)U(a,b): f(x)=1b−af(x) = \frac{1}{b-a} for a≤x≤ba \leq x \leq b. The exponential distribution: f(x)=λe−λxf(x) = \lambda e^{-\lambda x} for x≥0x \geq 0, with mean 1λ\frac{1}{\lambda}.
  • Student’s tt-distribution: Used when the population standard deviation is unknown and the sample size is small. t=xˉ−μs/nt = \frac{\bar{x} - \mu}{s / \sqrt{n}}. As n→∞n \to \infty, the tt-distribution approaches the normal distribution. Depends on degrees of freedom ν=n−1\nu = n - 1.
  • Chi-squared (χ2\chi^{2}) tests: Goodness of fit: tests whether observed frequencies match an expected distribution. χ2=∑(O−E)2E\chi^{2} = \sum \frac{(O - E)^{2}}{E}. Contingency tables: tests for independence between two categorical variables. Degrees of freedom: (rows−1)(columns−1)(\text{rows} - 1)(\text{columns} - 1).

Intuition

The PMF is like a bar chart of probabilities — each bar shows how likely a particular outcome is. The PDF is the continuous version: a curve where the area under the curve between two points gives the probability of falling in that range.

The tt-distribution is like the normal distribution but with heavier tails. This reflects the extra uncertainty from estimating σ\sigma with ss. With more data (larger nn), the tt-distribution converges to the normal because ss becomes a better estimate of σ\sigma.

Common Pitfalls

  1. Confusing PMF and PDF. A PMF gives probabilities directly: P(X=x)P(X = x). A PDF gives density: you must integrate to get probability. f(x)=0.3f(x) = 0.3 does NOT mean P(X=x)=0.3P(X = x) = 0.3 for a continuous variable.
  2. Forgetting that Var(aX+b)=a2Var(X)\text{Var}(aX + b) = a^{2}\text{Var}(X). The constant bb (shift) does not affect variance because variance measures spread, not location. A common error is writing Var(aX+b)=a Var(X)+b\text{Var}(aX + b) = a\,\text{Var}(X) + b.
  3. Using zz-test instead of tt-test. When σ\sigma is unknown and nn is small, use the tt-distribution, not the normal. Using zz gives confidence intervals that are too narrow.
  4. Miscounting degrees of freedom in χ2\chi^{2} tests. For goodness of fit, df=(number of categories)−1−(number of estimated parameters)df = (\text{number of categories}) - 1 - (\text{number of estimated parameters}). For contingency tables, df=(r−1)(c−1)df = (r-1)(c-1).

Cross-References