Statistics 2 - Poisson distributions

Question 1

What is the Poisson distribution?

Answer 1

If 𝑋~Po(λ), then the Poisson distribution is given by
P(𝑋=x)=(e(^-λ)λˣ)/𝑥!
Where 𝑋 is a discrete random variable.
The parameter, λ, in the Poisson distribution is the average number of times that the event will occur in a single interval.

Question 2

What are the necessary factors for the Poisson distribution to be a good model?

Answer 2

In order for the Poisson distribution to be a good model, the events must occur:

• independently
• singly, in space or time
• at a constant average rate so that the mean number in an interval is proportional to the length of the interval

Question 3

Describe the process of adding Poisson distribution.

Answer 3

If two Poisson variables 𝑋 and 𝑌 are independent, then the variable 𝑍=𝑋+𝑌 also has a Poisson distribution.
If 𝑋∼Po(λ₁) and 𝑌∼Po(λ₂), then (𝑋+𝑌)∼Po(λ₁+λ₂)

Question 4

How can it be proven that a random variable, X, has a Poisson distribution with parameter, λ?

Answer 4

If 𝑋∼Po(λ);
The mean of 𝑋=E(𝑋)=λ
The variance of 𝑋=Var(𝑋)=σ²=λ

Question 5

How are the mean and variance of a binomial random variable, X, calculated?

Answer 5

If 𝑋 is a binomial random variable wit 𝑋∼B(𝑛,𝑝), then:
the mean of 𝑋=E(𝑋)=μ=𝑛𝑝
the variance of 𝑋=Var(𝑋)=σ²=𝑛𝑝(1-𝑝)

Question 6

How can the Poisson distribution to approximate the binomial distribution and when would this approximation be useful and appropriate.

Answer 6

𝑋∼B(𝑛,𝑝) can be approximated with the Poisson distribution 𝑋∼Po(λ), where λ=𝑛𝑝, so long as:
𝑛 is large
𝑝 is small
There is no clear rule as to what constitutes ‘large 𝑛’ or ‘small 𝑝’ but usually, the value for 𝑛𝑝 will be ≤ 10.