State the probability function of X\simGeo(p).

P(X=x)=p(1-p)^{x-1} for x=1,2,3,\dots

List the conditions for a geometric model.

Two outcomes per trial, constant probability of success p, and independent trials.

What values can a geometric random variable take?

x=1,2,3,\dots with no upper limit.

Give P(Xx) for X\simGeo(p).

(1-p)^x, because the first x trials all fail.

A die is rolled until a six appears. Find the mean and variance.

X\simGeo\left(\frac16\right): mean 6, variance 30.

How do you solve (1-p)^n<k for n?

Take logarithms: n\frac{\ln k}{\ln(1-p)}, reversing the inequality because \ln(1-p)<0.

What is P(X odd) for X\simGeo(p)?

\frac{p}{1-(1-p)^2}=\frac{1}{2-p}, a geometric series with ratio (1-p)^2.

The geometric distributionEdexcel A-Level Further Maths: Flashcards

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What does a geometric random variable $X$ count?

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What does a geometric random variable $X$ count?: The number of independent trials up to and including the first success.
State the probability function of $X\sim\mathrm{Geo}(p)$ .: $\mathrm{P}(X=x)=p(1-p)^{x-1}$ for $x=1,2,3,\dots$
List the conditions for a geometric model.: Two outcomes per trial, constant probability of success $p$ , and independent trials.
What values can a geometric random variable take?: $x=1,2,3,\dots$ with no upper limit.
Give $\mathrm{P}(X>x)$ for $X\sim\mathrm{Geo}(p)$ .: $(1-p)^x$ , because the first $x$ trials all fail.
Give $\mathrm{P}(X\geq x)$ for $X\sim\mathrm{Geo}(p)$ .: $(1-p)^{x-1}$
Give $\mathrm{P}(X\leq x)$ for $X\sim\mathrm{Geo}(p)$ .: $1-(1-p)^x$
State $\mathrm{E}(X)$ for $X\sim\mathrm{Geo}(p)$ .: $\frac1p$
State $\mathrm{Var}(X)$ for $X\sim\mathrm{Geo}(p)$ .: $\frac{1-p}{p^2}$
A die is rolled until a six appears. Find the mean and variance.: $X\sim\mathrm{Geo}\left(\frac16\right)$ : mean $6$ , variance $30$ .
If $\mathrm{E}(X)=5$ for a geometric $X$ , what is $p$ ?: $p=\frac15=0.2$
How do you solve $(1-p)^n<k$ for $n$ ?: Take logarithms: $n>\frac{\ln k}{\ln(1-p)}$ , reversing the inequality because $\ln(1-p)<0$ .
What is $\mathrm{P}(X\text{ odd})$ for $X\sim\mathrm{Geo}(p)$ ?: $\frac{p}{1-(1-p)^2}=\frac{1}{2-p}$ , a geometric series with ratio $(1-p)^2$ .

Exam questions on The geometric distribution

A machine produces components, each of which is defective with probability $0.08$ , independently of all the others. The components are inspected one at a time. Let $X$ be the number of components inspected up to and including the first defective one.
Find the smallest number $n$ of inspections for which the probability that the first defective component has been found by the $n$ th inspection exceeds $0.5$ .2 marks
A driving test candidate passes at each attempt with probability $0.35$ , independently of any other attempt. Let $X$ be the number of attempts the candidate makes up to and including the first one that they pass.
Find the probability that the candidate needs at least $5$ attempts to pass, and the expected number of attempts needed.2 marks
An archer hits the target on each shot with probability $p$ , independently of all other shots. Let $X$ be the number of shots up to and including the first hit. It is known that $\mathrm{E}(X)=5$ .
Find the value of $p$ , and the variance and standard deviation of $X$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).