What does X\simPo(\lambda) mean?

X is the number of events in an interval, with mean \lambda.

Poisson probability formula?

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

Why might a Poisson model fail for calls over a whole day?

The average rate is not constant (busy and quiet periods).

Sample mean 10 but sample variance 30: what does it suggest?

The data are not Poisson, because the mean and variance should be about equal.

Poisson model and probabilitiesAQA A-Level Further Maths: Flashcards

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State the conditions for a Poisson model.

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State the conditions for a Poisson model.: Events occur singly, at random, independently, at a constant average rate.
What does $X\sim\mathrm{Po}(\lambda)$ mean?: $X$ is the number of events in an interval, with mean $\lambda$ .
Poisson probability formula?: $\mathrm{P}(X=r)=\frac{e^{-\lambda}\lambda^r}{r!}$ , $r=0,1,2,\dots$
Mean of $\mathrm{Po}(\lambda)$ ?: $\lambda$
Variance of $\mathrm{Po}(\lambda)$ ?: $\lambda$ , equal to the mean.
Standard deviation of $\mathrm{Po}(\lambda)$ ?: $\sqrt{\lambda}$
How do you find $\mathrm{P}(X\ge4)$ ?: $1-\mathrm{P}(X\le3)$
How do you find $\mathrm{P}(3\le X\le6)$ ?: $\mathrm{P}(X\le6)-\mathrm{P}(X\le2)$
2.5 faults per metre: what is $\lambda$ for 4 m?: $\lambda=10$
$\mathrm{P}(X=0)$ for $\mathrm{Po}(\lambda)$ ?: $e^{-\lambda}$
If $\mathrm{P}(X=0)=0.2$ , what is $\lambda$ ?: $\lambda=-\ln0.2=\ln5=1.61$
Why might a Poisson model fail for calls over a whole day?: The average rate is not constant (busy and quiet periods).
Sample mean 10 but sample variance 30: what does it suggest?: The data are not Poisson, because the mean and variance should be about equal.
$\mathrm{P}(X=2)$ for $\mathrm{Po}(2.5)$ ?: $0.2565$

Exam questions on Poisson model and probabilities

Faults occur at random, independently and at a constant average rate of 2.5 per metre in a long roll of fabric. The number of faults, $X$ , in one metre is modelled by $X\sim\mathrm{Po}(2.5)$ .
Find the probability that a 2 metre length of the fabric contains no faults.2 marks
A call centre receives calls at random at a constant average rate of 6 per hour. The number of calls, $Y$ , received in one hour is modelled by $Y\sim\mathrm{Po}(6)$ .
The manager notices that calls are much more frequent between 12 noon and 1 pm than at other times of the day. Explain why $\mathrm{Po}(6)$ is not suitable as a model for the number of calls in every hour of the working day.2 marks
The number of typing errors on a page of a manuscript is modelled by $X\sim\mathrm{Po}(1.8)$ . Errors on different pages occur independently of one another.
Find $\mathrm{P}(X\ge3)$ .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).