What three conditions must hold for a Poisson model to be appropriate?

The events must occur independently , singly , and at a constant average rate . Occurring singly means that no two events happen at exactly the same moment, or at exactly the same point in space.

People arriving at a restaurant are modelled by a Poisson distribution. How might each of the three conditions fail?

They may not arrive independently , since people invite others they know; they may not arrive singly , since they come in groups; and the rate may not be constant , being higher at dinner time than in the afternoon. Criticising a Poisson model means testing the conditions against the actual context, not against the numbers.

True or False? A Poisson random variable has no largest possible value.

True. A Poisson variable can take any whole number from 0 upwards, so there is no maximum. The probabilities do get closer and closer to zero as the value grows, which is what allows them all to sum to 1.

What must be true of two Poisson variables before you can add them?

They must be independent , and they must model events over the same interval . They need not model the same kind of event: pedestrians and cyclists passing a window can be added, provided both rates are given for the same length of time.

A LevelFurther MathsEdexcelFurther Statistics 1FlashcardsDiscrete Probability DistributionsPoisson & Binomial Distributions

Poisson & Binomial Distributions (Edexcel A Level Further Maths: Further Statistics 1): Flashcards

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FrontThe Poisson Distribution
Define the Poisson distribution.

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Cards in this collection (16)

Define the Poisson distribution.
The Poisson distribution $Po (λ)$ models the number of times an event occurs within a fixed interval of time or space.
The parameter $λ$ is the average number of occurrences in one such interval, and $X ~ Po (λ)$ counts the occurrences.
What three conditions must hold for a Poisson model to be appropriate?
The events must occur independently, singly, and at a constant average rate.
Occurring singly means that no two events happen at exactly the same moment, or at exactly the same point in space.
For a Poisson variable, complete the two identities that turn strict inequalities into weak ones:
$P (X < x) = P (X \leq______) and P (X > x) = P (X \geq______)$
The completed identities are:
$P (X < x) = P (X \leq x - 1) and P (X > x) = P (X \geq x + 1)$
They work because a Poisson variable takes only whole number values, so excluding $x$ itself simply moves the boundary along by one.
People arriving at a restaurant are modelled by a Poisson distribution. How might each of the three conditions fail?
They may not arrive independently, since people invite others they know; they may not arrive singly, since they come in groups; and the rate may not be constant, being higher at dinner time than in the afternoon.
Criticising a Poisson model means testing the conditions against the actual context, not against the numbers.
True or False?
A Poisson random variable has no largest possible value.
True.
A Poisson variable can take any whole number from 0 upwards, so there is no maximum.
The probabilities do get closer and closer to zero as the value grows, which is what allows them all to sum to 1.
The number of texts received in an hour is modelled by $Po (λ)$ . What models the number received in 15 minutes?
The distribution is $Po (\frac{λ}{4})$ , because the parameter scales by exactly the same factor as the interval.
This works for intervals in space as well as in time, and it is the constant average rate condition that makes the scaling valid.
For independent $X ~ Po (λ)$ and $Y ~ Po (μ)$ , complete the distribution of their sum:
$X + Y ~ Po (______)$
The completed result is:
$X + Y ~ Po (λ + μ)$
The sum of two independent Poisson variables is itself Poisson, and its mean is simply the sum of the two separate means.
What must be true of two Poisson variables before you can add them?
They must be independent, and they must model events over the same interval.
They need not model the same kind of event: pedestrians and cyclists passing a window can be added, provided both rates are given for the same length of time.
The mean and variance of a sample are calculated. How does that help you judge whether a Poisson model is suitable?
If they are approximately equal a Poisson model may be suitable, because the mean and the variance of a Poisson distribution are both $λ$ .
If they are clearly different the model cannot be appropriate, though equality on its own is not enough, since the three conditions must hold as well.
How do you rewrite $P (X \geq 4)$ using a cumulative probability?
Write it as $1 - P (X \leq 3)$ , since the values 4, 5, 6 and upwards are everything except 0, 1, 2 and 3.
Listing the integers is the reliable way to fix the boundary, because subtracting $P (X \leq 4)$ by mistake is easily done.
When can a binomial distribution be approximated by a Poisson distribution?
When $n$ is large and $p$ is small, with $n > 30$ a useful guide and $n p$ usually no more than 10.
There is no firm rule for how large or how small they have to be, so those figures are guidance rather than a test to be passed.
Complete the parameter to use when approximating $X ~ B (n, p)$ by a Poisson distribution:
$λ =______$
The completed parameter is:
$λ = n p$
This gives the approximating Poisson distribution exactly the same mean as the binomial it is replacing.
$n$ is large but $p$ is close to 1, so a Poisson approximation does not apply directly. What can you do?
Model the number of failures instead, using $X' ~ B (n, 1 - p)$ , where $1 - p$ is small.
A Poisson approximation can then be used on $X'$ , and the answer converted back into the question's own terms at the end.
True or False?
A Poisson approximation to a binomial distribution has the same variance as the binomial.
False.
Only the means are matched, at $n p$ : the binomial variance is $n p (1 - p)$ while the Poisson variance is $n p$ .
The two are close only when $p$ is small enough for $1 - p$ to be near 1, which is precisely why the approximation requires a small $p$ .
How is the Poisson distribution related to the binomial distribution?
The Poisson distribution is derived from the binomial by letting $n$ become infinitely large and $p$ infinitely small.
That is why a binomial distribution with a large $n$ and a small $p$ is well approximated by a Poisson one.
One visitor in a thousand to a website subscribes, and the site receives 3000 visits. Which Poisson distribution approximates the number of subscribers?
The approximating distribution is $Po (3)$ , because $3000 \times 0.001 = 3$ .
Here $n$ is large and $p$ is very small, so the conditions for the approximation are comfortably met.

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