(R) The Poisson Distribution

In this article, we will be discussing The Poisson distribution. The Poisson distribution is a discrete distribution, and is similar to the binomial distribution. However, the Poisson distribution applies to occurrences over a specified interval. The random variable 'x' is the number of occurrences of the event in an interval.

μ or λ = (Mean or Lambda) The average number of occurrences of the event over the interval.

x = The number of occurrences of the event over the interval.

Requirements

The random variable 'x' is the number of occurrences of an event over some interval.

The occurrences must be random.

The occurrences must be independent from each other.

The occurrences must be uniformly distributed over the interval being used (cannot be seasonal).

Differences from a Binomial Distribution

The Possion distribution differs from the binomial distribution in these fundamental ways:

The binomial distribution is affected by the sample size 'n' and the probability 'p', whereas the Poisson distribution is affected only by the mean.

In a binomial distribution the possible values of the random variable x are 0,1,...,n, but a Poisson distribution has possible 'x' values of 0,1,2..., WITH NO UPPER LIMIT! (emphasis added)

* Source for the above material: https://www.youtube.com/watch?v=BR1nN8DW2Vg 

   User: DrCraigMcBridePhd Video: "Statistics - Binomial & Poisson Distributions" 

Example in R:

Let's say, that every Tuesday in spring, for 8 hours, your web camera films for blue birds which frequent your garden. You have counted these blue birds, and over the course of the season, have noticed an average of 12 blue birds visiting your garden each day.

What is the probability that EXACTLY 8 blue birds will visit your garden given the parameters of the experiment?

P(x) = 8

λ = 12

In R, this would be expressed in the code below:

dpois(x=8, lambda=12)

Which would generate the result of:

[1] 0.06552328

What is the probability that exactly 0 blue birds will visit?
What is the probability that exactly 1 blue bird will visit?
What is the probability that exactly 2 blue birds will visit?
What is the probability that exactly 3 blue birds will visit?

In R, this would be expressed in the code below:

dpois(x=0:3, lambda=12)

Which would generate the result of:

[1] 6.144212e-06 7.373055e-05 4.423833e-04 1.769533e-03

What is the probability that 6 or less blue birds will visit your garden?

P(x <= 6)

λ = 12

In R, this would be expressed in the code below:

sum(dpois(x=0:6, lambda=12))

or

ppois(q=6, lambda=12, lower.tail=T)

Either, which would generate the result of:

[1] 0.04582231

What is the probability that more than 6 blue birds will visit your garden?

P(x  > 6) 

λ = 12

In R, this would be expressed in the code below:

ppois(q=6, lambda=12, lower.tail=F)

Which would generate the result of:

[1] 0.9541777

Alternatively, you could also utilize any of the following lines of code and achieve the same result:

1 - sum(dpois(x=0:6, lambda=12))

sum(dpois(x=138, lambda=12))

In the next article we will discuss the normal distribution. I hope to see you then. All the best, data monkeys.