Binomial and Poisson Distribution in R – Explore the complete concept!

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In this article, we are going to cover what is Binomial and Poisson Distribution in R. Along with this, we will study various uses of it, other symbols, formulas, and their differences.

So, let’s start the tutorial.

Binomial Distribution in R

It is applied to a single variable discrete data where results are the no. of “successful outcomes”.

For example:

  • The number of times the lights are red in 10 sets of traffic lights,
  • Number of students with green eyes in a class of 30,
  • A number of plants with diseased leaves from a sample of 60 plants.

We use it to calculate the probability of occurrences exactly, less than, more than, between given values.

For example:

  • “ The probability that the number of red lights will be exactly 7”
  • “Probability that the number of green-eyed students will be less than 5”
  • “The probability that the no. of diseased plants will be more than 20”

Don’t forget to check the tutorial on Classification in R

1. Parameters, Statistics, and symbols involved in Binomial Distribution

Symbol Population Parameter Symbol
  Sample Statistic
The Probability of Successπp
Sample SizeNn

2. Other Symbols

  • X – The no. of successful outcomes wanted.
  • ⁿCr – The number of ways in which x “successes” can be chosen from sample size n. We use ⁿCr key on our calculator in the formula.

Formula used:


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where, x = No. of success.
(n-x) = No. of failures.
(1-p) = Probability of failure.
p^x = Probability of success.

Poisson Distribution in R

We call it the distribution of rare events., a Poisson process is where DISCRETE events occur in a continuous, but finite interval of time or space in R.

The following conditions must apply:

  • For a small interval, the probability of the event occurring is proportional to the size of the interval.
  • The probability of more than one occurrence in the small interval is negligible.
  • Each occurrence must be independent of others and must be at random.
  • The events are often defects, accidents or unusual natural happenings, such as an earthquake.
  • The parameter for the Poisson distribution is a lambda. It is average or mean of occurrences over a given interval.
  • The probability function is: for x= 0,1.2,3 ….

You must have a look at the Clustering in R Programming

Difference between Binomial and Poisson Distribution in R

Binomial Distribution:

  • Fixed no. of Trials (n) [10 pie throws], although, only two possible outcomes are possible.
  • A probability of success is constant(p).
  • Each trial is independent.
  • Also, it predicts no.s of successes within a set no. of trials.
  • We use it to test for independence.

Poisson Distribution

  • Infinite no. of trials.
  • Also, it has unlimited no. of outcomes possible.
  • The mean of the distribution is the same for all intervals.
  • No. of occurrence in any given interval independent of others.
  • Also, it predicts no. of occurrences per unit, time, space.
  • We use it to test for independence.


We have discussed Binomial and Poisson distributions in R. Also, we have covered their usages, symbols, and the difference between Binomial and Poisson distribution.

Now, you must learn about Normal Distribution in R Programming

Still, if you have any query in R Binomial and R Poisson Distribution, ask in the comment section.

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