State the probability function of X\simPo(\lambda).

P(X=r)=e^{-\lambda}\frac{\lambda^r}{r!} for r=0,1,2,\ldots

State three conditions for a Poisson model.

Events occur independently, singly, and at a constant average rate.

State the additive property.

If X\simPo(\lambda) and Y\simPo(\mu) are independent, then X+Y\simPo(\lambda+\mu).

What does it suggest if the sample variance is much larger than the sample mean?

The events are clustered, so a Poisson model is not appropriate.

Give a reason why buses may not be modelled by a Poisson distribution.

They run to a timetable, so arrivals are not random or independent.

What does the parameter \lambda represent?

The mean number of events in the interval being considered.

The Poisson distributionEdexcel International A Level Further Maths: Flashcards

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State the probability function of $X\sim\text{Po}(\lambda)$.

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State the probability function of $X\sim\text{Po}(\lambda)$ .: $P(X=r)=e^{-\lambda}\frac{\lambda^r}{r!}$ for $r=0,1,2,\ldots$
What are $E(X)$ and $\text{Var}(X)$ for $X\sim\text{Po}(\lambda)$ ?: Both equal $\lambda$ .
State three conditions for a Poisson model.: Events occur independently, singly, and at a constant average rate.
Rewrite $P(X\ge5)$ for use with tables.: $1-P(X\le4)$
Rewrite $P(X<5)$ for use with tables.: $P(X\le4)$
Rewrite $P(3\le X\le6)$ for use with tables.: $P(X\le6)-P(X\le2)$
Rate 3 per minute. What is the distribution of the number of events in 5 minutes?: $\text{Po}(15)$
State the additive property.: If $X\sim\text{Po}(\lambda)$ and $Y\sim\text{Po}(\mu)$ are independent, then $X+Y\sim\text{Po}(\lambda+\mu)$ .
What does it suggest if the sample variance is much larger than the sample mean?: The events are clustered, so a Poisson model is not appropriate.
Give a reason why buses may not be modelled by a Poisson distribution.: They run to a timetable, so arrivals are not random or independent.
$X\sim\text{Po}(2)$ . Write down $P(X=0)$ .: $e^{-2}=0.135$
What does the parameter $\lambda$ represent?: The mean number of events in the interval being considered.

Exam questions on The Poisson distribution

Typing errors occur at random in a manuscript at a mean rate of 1.5 per page. The number of errors on one page is modelled by $X\sim\text{Po}(1.5)$ .
Find the probability that a 2-page section contains no errors.2 marks
A call centre receives calls at random at a constant mean rate of 4 calls per 10 minutes.
Find the probability that more than 6 calls are received in a 10-minute period.2 marks
Flaws occur at random in a roll of fabric at a mean rate of 0.4 per square metre. Flaws in separate pieces of fabric occur independently.
Find the probability that a 5 m $^2$ piece of fabric contains at most 3 flaws.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).