S4 Statistical distributions — 9MA0 spec coverage

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.

A discrete probability distribution lists possible values with probabilities between $0$ and $1$ whose total is $1$ ; a discrete uniform distribution assigns equal probability to each value.
Use $X\sim\operatorname{B}(n,p)$ for a fixed number $n$ of independent trials, each with two outcomes and constant success probability $p$ .
For a binomial variable, $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ ; cumulative probabilities are often most efficiently found with a calculator.
Translate inequalities carefully: for integer-valued $X$ , $P(X<r)=P(X\leq r-1)$ and $P(X\geq r)=1-P(X\leq r-1)$ .

Tier 1 · Easy

3 marks

ORIGINAL

A discrete random variable $X$ takes values $0$ , $1$ and $2$ with probabilities $k$ , $3k$ and $4k$ respectively. Find $k$ and $P(X\geq1)$ .

Tier 2 · Standard

3 marks

ORIGINAL

Let $X\sim\operatorname{B}(14,0.4)$ . Find $P(X\leq4)$ to $4$ decimal places.

Tier 3 · Hard

5 marks

ORIGINAL

A binomial random variable $X$ has $10$ trials and satisfies $P(X=0)=0.8^{10}$ . Find its success probability $p$ , then calculate $P(X\geq3)$ to $4$ decimal places.

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 $X\sim\operatorname{N}(\mu,\sigma^2)$ , where $\mu$ is the mean and $\sigma$ is the standard deviation; standardise with $Z=\frac{X-\mu}{\sigma}$ .
The Normal curve is continuous, symmetric about $\mu$ , has total area $1$ , and has points of inflection at $\mu-\sigma$ and $\mu+\sigma$ .
A Normal model is plausible for a roughly symmetric, unimodal histogram with no strong outliers, but context and the variable's possible values also matter.
A binomial distribution may be approximated by a Normal distribution when $n$ is large and $p$ is close to $0.5$ ; use a continuity correction.

Tier 1 · Easy

2 marks

ORIGINAL

A random variable has distribution $X\sim\operatorname{N}(50,6^2)$ . Find $P(X<56)$ to $4$ decimal places.

Tier 2 · Standard

4 marks

ORIGINAL

The lifetime $L$ of a component, in hours, is modelled by $L\sim\operatorname{N}(72,8^2)$ . Find the lifetime exceeded by exactly $10\%$ of components, to $1$ decimal place.

Tier 3 · Hard

5 marks

ORIGINAL

Let $X\sim\operatorname{B}(200,0.35)$ . Use a Normal approximation with a continuity correction to estimate $P(X\geq82)$ , giving your answer to $4$ decimal places.

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.

Tier 1 · Easy

2 marks

ORIGINAL

A manufacturer inspects $12$ independently chosen switches. Each switch has probability $0.04$ of being faulty. State a suitable distribution for the number $F$ of faulty switches and give its parameters.

Tier 2 · Standard

3 marks

ORIGINAL

The masses of loaves from a stable production line form a roughly symmetric, single-peaked histogram with no clear outliers. Explain why a Normal model may be suitable and why a binomial model is not.

Tier 3 · Hard

5 marks

ORIGINAL

A technician proposes $D\sim\operatorname{B}(500,p)$ for the number of defective pixels on each screen. Defects tend to occur in neighbouring clusters, and $p$ varies between production shifts. Critique the model and suggest how the data should be used before choosing a replacement.