Before choosing a Poisson model for some data, what quick check can you make?

Work out the mean and the variance of the data and see whether they come out roughly equal . A Poisson distribution has them exactly equal, so data whose variance is far from its mean is telling you the model does not fit. It is only a check, not a proof: the three conditions still have to be considered.

A LevelMathsCambridge (CIE)Probability & Statistics 2FlashcardsStatistical DistributionsPoisson Distribution

Poisson Distribution (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Flashcards

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FrontThe Poisson Distribution
Define Poisson distribution.

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Define Poisson distribution.
The Poisson distribution models a discrete random variable that counts the number of events occurring at random in a given interval of time or space.
It is written $X ~ Po (λ)$ , where $λ$ is the mean number of occurrences in that interval.
Unlike a binomial there is no fixed number of trials, so $X$ can in principle be any non-negative whole number.
Fill in the three conditions a Poisson model requires:
events occur $______$ and at random in the interval
the mean number of occurrences, $λ$ , is known and $______$
each occurrence is $______$ of the others
The completed conditions are:
events occur singly and at random in the interval
the mean number of occurrences, $λ$ , is known and finite
each occurrence is independent of the others
Singly means two events cannot happen at exactly the same instant, and $λ$ must be positive but need not be a whole number.
For $X ~ Po (λ)$ , what are $E (X)$ and $Var (X)$ ?
They are both exactly $λ$ .
The mean and the variance of a Poisson distribution are equal, which no other distribution on this course does.
The standard deviation is therefore $\sqrt{λ}$ , and there is only ever one parameter to find.
How does the shape of a Poisson distribution depend on $λ$ ?
A small $λ$ gives a strong tail to the right, with most of the probability piled up at 0 and 1.
As $λ$ grows the distribution becomes steadily more symmetrical, and by about $λ = 5$ it is already roughly so.
That drift towards symmetry is exactly why a large $λ$ can be approximated by a normal distribution.
Complete the formula for the probability of exactly $r$ occurrences:
$P (X = r) = e^{- λ} \times \frac{λ^{r}}{______}$
The completed formula is:
$P (X = r) = e^{- λ} \times \frac{λ^{r}}{r!}$
It applies for $r = 0, 1, 2, \dots$ with no upper limit, which is what distinguishes it from a binomial.
Any non-integer or negative value of $r$ has probability zero.
Before choosing a Poisson model for some data, what quick check can you make?
Work out the mean and the variance of the data and see whether they come out roughly equal.
A Poisson distribution has them exactly equal, so data whose variance is far from its mean is telling you the model does not fit.
It is only a check, not a proof: the three conditions still have to be considered.
What must you state before writing down a Poisson model?
What your random variable is, in words, for example "let $X$ be the number of typing errors per page".
The interval matters as much as the count, since $λ$ is a mean per interval and is meaningless without one.
Only then write $X ~ Po (λ)$ with the value of $λ$ that belongs to that interval.
A team scores a mean of 2 goals an hour. What distribution models the goals in a 90-minute game?
It is $Po (3)$ , because 90 minutes is $1.5$ hours and $λ$ scales with the interval.
So multiply the given mean by however many of the stated intervals the question asks about.
This is worth checking at every part of a question, because the interval often changes between parts.
How do you find $P (X \leq 3)$ for a Poisson distribution without typing four separate terms?
Every term contains $e^{- λ}$ , so factorise it out:
$P (X \leq 3) = e^{- λ} (1 + λ + \frac{λ^{2}}{2!} + \frac{λ^{3}}{3!})$
The first two terms simplify because $λ^{0} = 1$ and $0! = 1! = 1$ .
That single bracket is far quicker to type and much less prone to slips than four separate products.
True or False?
For a Poisson distribution, subtracting from 1 is merely a quicker way of finding $P (X > k)$ .
False.
It is the only way. A Poisson variable has no upper limit, so there are infinitely many values greater than $k$ and they cannot be added up directly.
So $P (X > k) = 1 - P (X \leq k)$ is a necessity here, where for a binomial it would only have been a convenience.

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