Model situations with the binomial and Poisson distributions, state the conditions for each, and compute probabilities, means and variances

What this dot point is asking

SEAB wants you to recognise when the binomial and Poisson distributions apply, state the conditions for each, and compute probabilities, means and variances, as well as choose the correct model for a given situation.

The answer

The binomial distribution

$X \sim \mathrm{B}(n, p)$ models the number of successes in $n$ independent trials, each with constant success probability $p$. The conditions: a fixed number of trials, two outcomes per trial, independent trials, and constant $p$. The probability function is

Its mean and variance are $\mathrm{E}(X) = np$ and $\operatorname{Var}(X) = np(1 - p)$.

The Poisson distribution

$X \sim \mathrm{Po}(\lambda)$ models the number of events in a fixed interval when events occur independently at a constant average rate $\lambda$, with no fixed upper limit. The probability function is

A defining feature: the mean and variance are equal, $\mathrm{E}(X) = \operatorname{Var}(X) = \lambda$.

Choosing the model

• Use the binomial when there is a fixed number of trials and you count successes.
• Use the Poisson when you count occurrences over an interval of time or space with a known average rate and no natural maximum.

The Poisson is also the limit of the binomial when $n$ is large and $p$ small with $np = \lambda$ moderate.

Combining Poisson variables

If $X \sim \mathrm{Po}(\lambda_1)$ and $Y \sim \mathrm{Po}(\lambda_2)$ are independent, then $X + Y \sim \mathrm{Po}(\lambda_1 + \lambda_2)$: rates over combined intervals add.

Examples in context

Example 1. Exam multiple choice. The number correct by pure guessing on $30$ four-option questions is $\mathrm{B}(30, 0.25)$, with expected score $30 \times 0.25 = 7.5$, the baseline against which real performance is judged.

Example 2. Rare faults on a cable. Flaws occurring randomly along a cable at $0.2$ per metre follow a Poisson model; the probability of a flawless $10$ m length uses $\lambda = 2$ and $\mathrm{P}(X = 0) = e^{-2}$.

Try this

Q1. State the mean and variance of $\mathrm{B}(50, 0.2)$. [2 marks]

• Cue. Mean $= 10$, variance $= 50(0.2)(0.8) = 8$.

Q2. For $X \sim \mathrm{Po}(4)$, find $\mathrm{P}(X = 0)$. [2 marks]

• Cue. $e^{-4} \approx 0.0183$.

Q3. State one condition distinguishing when to use the Poisson rather than the binomial. [1 mark]

• Cue. Events occur over an interval with a known rate and no fixed maximum number of trials.