What is the probability function of X\simPo(\lambda)?

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

What does \lambda represent?

The mean number of events in the interval.

State three conditions for a Poisson model.

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

What is the additive property of Poisson distributions?

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

What must be true for the additive property to hold?

The two variables must be independent.

Give a context where independence fails for vehicle counts.

Convoys behind slow vehicles: one vehicle passing makes others likely to follow.

How does a Poisson model differ from a binomial model?

No fixed number of trials; the count has no upper limit.

The Poisson distributionEdexcel A-Level Further Maths: Flashcards

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What is the probability function of $X\sim\text{Po}(\lambda)$?

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What is the probability function of $X\sim\text{Po}(\lambda)$ ?: $P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}$ , for $x=0,1,2,\ldots$
What does $\lambda$ represent?: The mean number of events in the interval.
State three conditions for a Poisson model.: Events occur singly, independently, and at a constant mean rate.
How do you find $P(X\geq k)$ for a Poisson variable?: $1-P(X\leq k-1)$
Write $P(2<X\leq6)$ using cumulative probabilities.: $P(X\leq6)-P(X\leq2)$
6 events per hour. What is the distribution for 20 minutes?: $\text{Po}(2)$
If $X\sim\text{Po}(\lambda)$ per minute, what is the distribution for 5 minutes?: $\text{Po}(5\lambda)$
What is the additive property of Poisson distributions?: If $X\sim\text{Po}(\lambda)$ and $Y\sim\text{Po}(\mu)$ are independent, then $X+Y\sim\text{Po}(\lambda+\mu)$ .
What must be true for the additive property to hold?: The two variables must be independent.
Give a context where independence fails for vehicle counts.: Convoys behind slow vehicles: one vehicle passing makes others likely to follow.
What is $P(X=0)$ for $X\sim\text{Po}(\lambda)$ ?: $e^{-\lambda}$
How does a Poisson model differ from a binomial model?: No fixed number of trials; the count has no upper limit.

Exam questions on The Poisson distribution

The number of emails, $X$ , received by an office in a 10-minute period is modelled by $X\sim\text{Po}(3.5)$ .
Find $P(2<X\leq6)$ .2 marks
A helpline receives calls at random, independently of each other, at a mean rate of 6 per hour.
Find the probability that exactly one call is received in each of two successive 20-minute periods.2 marks
A café receives online orders at a mean rate of 3 per hour and phone orders at a mean rate of 2 per hour. Both types of order arrive at random and independently of each other.
Find the probability that at least 2 orders in total are received in a 30-minute period.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).