Discrete random variables and probability distributions, the binomial distribution as a model and its probabilities, the Normal distribution, standardising, the inverse Normal, and the Normal approximation to the binomial.

What this dot point is asking

OCR wants you to work with discrete random variables and their probability distributions (probabilities summing to one, and the mean), use the binomial distribution as a model (knowing its conditions) and find binomial probabilities, model continuous data with the Normal distribution, standardise to the standard Normal to find probabilities, use the inverse Normal to find unknown values or parameters, and apply the Normal approximation to the binomial.

The answer

Discrete random variables

A discrete random variable takes separate values, each with a probability; the probabilities must sum to $1$. The mean (expected value) is $E(X) = \sum x\,P(X = x)$, the long-run average value.

The binomial distribution

The binomial $X \sim B(n, p)$ counts the successes in $n$ independent trials, each with the same success probability $p$.

For a cumulative probability, use the complement to avoid long sums: $P(X \ge r) = 1 - P(X \le r - 1)$.

The Normal distribution and standardising

The Normal distribution $X \sim N(\mu, \sigma^2)$ models continuous, symmetric, bell-shaped data. Convert to the standard Normal $Z \sim N(0, 1)$ to find probabilities.

The inverse Normal

When a probability is given and a value is wanted, work backwards: find the $z$-value for that probability (the inverse Normal), then convert to $X$ with $X = \mu + z\sigma$. Two such conditions give simultaneous equations for an unknown $\mu$ and $\sigma$.

Examples in context

Choosing a binomial model

Before using the binomial, check the conditions hold. A fixed number of independent trials with a constant success probability fits; sampling without replacement from a small population does not, because $p$ changes between trials.

The Normal approximation to the binomial

When $n$ is large and $p$ is not too close to $0$ or $1$, the binomial $B(n, p)$ is approximately Normal with the same mean and variance, $N(np, np(1 - p))$. Because you replace a discrete distribution with a continuous one, apply a continuity correction (for example $P(X \ge 50)$ becomes $P(X > 49.5)$).

Try this

Q1. $X \sim B(8, 0.25)$. Find the mean and variance. [2 marks]

• Cue. Mean $= 8(0.25) = 2$; variance $= 8(0.25)(0.75) = 1.5$.

Q2. $X \sim N(50, 16)$. Find $P(X < 54)$. [2 marks]

• Cue. $Z = \dfrac{54 - 50}{4} = 1$, so $P(Z < 1) \approx 0.8413$.