S4 Statistical distributions — notes on one page

S4.1 · Understand and use simple, discrete probability distributions (mean and variance of discrete random variables excluded), including the binomial distribution as a model; calculate probabilities using the binomial distribution.

Definition. Specification S4.1: understand and use simple, discrete probability distributions (calculation of the mean and variance of discrete random variables is excluded), including the binomial distribution as a model; calculate probabilities using the binomial distribution.

• A discrete distribution lists the values with probabilities that are each between $0$ and $1$ and add to $1$; a discrete uniform distribution gives every value the same probability.
• $X\sim\mathrm B(n,p)$ models the number of successes in $n$ trials when the trials are independent, each has two outcomes and the probability of success $p$ is constant; those conditions are the answer to “explain why the binomial is suitable”.
• Write the model down, $X\sim\mathrm B(10,\tfrac16)$, before any calculation.
• Point probabilities use $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ and cumulative probabilities come from the calculator, but the inequality must be translated for an integer variable: $P(X<3)=P(X\leq2)$, $P(X\geq3)=1-P(X\leq2)$, $P(X>3)=1-P(X\leq3)$.
• A probability found in one part often becomes the $p$ of a second binomial in the next, and $np$ gives an expected count.

What the examiner wants. The inequality read correctly. The June 2024 Paper 31 report says almost all wrote down the right binomial model and found $P(X=3)$, but for $P(X<3)$ an answer of $0.930$ from $P(X\leq3)$ was often seen, suggesting unfamiliarity with the calculator, and others misread the inequality and calculated $1-P(X\leq2)$.

Exam tip. Translate the inequality into the exact binomial range before using cumulative probabilities and round only at the end.

S4.2 · Understand and use the Normal distribution as a model; find probabilities using the Normal distribution; link to histograms, mean, standard deviation, points of inflection and the binomial distribution.

Definition. Specification S4.2: understand and use the Normal distribution as a model; find probabilities using the Normal distribution; link to histograms, mean, standard deviation, points of inflection and the binomial distribution.

• Write the model as $X\sim\mathrm N(\mu,\sigma^2)$, with the variance, not the standard deviation, in the bracket.
• The curve is symmetric about $\mu$ with points of inflection at $\mu\pm\sigma$, so a bell-shaped histogram and a mean close to the median support a Normal model.
• Probabilities come from the calculator, but the working must show the standardisation $z=\dfrac{x-\mu}{\sigma}$ when $\mu$ or $\sigma$ is unknown: an inverse-Normal value such as $1.2816$ for the top $10\%$ gives one linear equation in $\mu$ and $\sigma$, and two such equations are solved simultaneously.
• Keep the same units throughout.
• A Normal variable is continuous, so there is no continuity correction; the correction, $P(X>95.5)$ for “more than $95$”, belongs only to approximating a binomial $\mathrm B(n,p)$ by $\mathrm N(np,\,np(1-p))$ when $n$ is large and $p$ is near $0.5$.

What the examiner wants. A Normal used as a continuous model in the right units. The June 2024 Paper 31 report says errors in finding $P(T<300)$ came from applying a continuity correction, $P(T<299.5)$, or from not converting minutes to seconds, while the strongest answers to the unknown-parameter part wrote down a pair of linear equations in $\mu$ and $\sigma$ with suitable $z$-values.

Exam tip. Convert an exceedance percentage to the corresponding lower-tail probability before standardising or using an inverse Normal calculation.

S4.3 · Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate.

• Select a distribution by matching its assumptions to the variable and data-generating process, not merely because its parameters can be estimated.
• A binomial model needs a fixed number of trials, two outcomes per trial, independence and a constant success probability.
• A Normal model is continuous and symmetric with unbounded tails, so it can be unsuitable for strongly skewed, bounded or discrete data.
• Support a choice with contextual evidence such as histogram shape, stability over time and dependence; state how a failed assumption could affect predictions.

Exam tip. Justify a model using the variable type, distribution shape and process assumptions, and name any condition that may fail.