**Binomial Distribution vs Poisson Distribution**

The **main difference between Binomial and Poisson Distribution **is that the Binomial distribution is only for a certain frame or a probability of success and the Poisson distribution is used for events that could occur a very large number of times.

There are many types of a theorem like a normal theorem, Gaussian Distribution, Binomial Distribution, Poisson Distribution and many more to get the probability of an event.

The probability distribution is based on the probability theory to explain the random variable’s behavior.

Bernoulli distribution is a two-fold experiment with fixed p and 1−p probabilities at another side with the Poisson distribution we can get a probability of a continuous event or a variable.

Parameter of Comparison | Binomial Distribution | Poisson Distribution |
---|---|---|

Possibility of Outcome | There are only 2 outcomes. i.e.:- Yes or No / True or False | There are no limit of possible outcome. |

Success | The probability of success is constant | Infinitesimal ability to succeed |

Nature | It’s with Biparamatric nature | It’s with Uniparametric nature |

Mean and Variable | Mean > Variable | Mean = Variable |

Example | A Dice Experiment | Mistake in speaking in a year |

## What is Binomial Distribution?

A Swiss mathematician Jakob Bernoulli had derived the formula of the Binomial Distribution.

**In an experiment or test that is repeated several times, a binomial distribution can be viewed as simply the possibility of a SUCCESS or FAILURE outcome.** The binomial is a form of distribution with two possible results**. **

**Formula := b(x; n, P) = _{n}C_{x} * P^{x} * (1 – P)^{n – x}**

^{}Where n is the total number of trials of the event. P is the probability for success in individual trial and the x is the number of successes that result from the binomial experiment.

**Conditions to apply Binomial Distribution:-**

*The number of observations n is fixed.**Each observation is independent.**Each observation represents one of two outcomes (“success” or “failure”).**The probability of “success” p is the same for each outcome.*

**Examples:**

- The number of times the lights are green in 10 sets of traffic lights,
- Number of teachers with uniforms in the staffroom of 30,
- A number of plants with diseased leaves from a sample of 60 plants.

## What is Poisson Distribution?

This distribution was invented by the famous French mathematician Simon Denis Poisson. This is often referred to as the distribution of rare events.

**A Poisson cycle wherein continues but a finite interval of time or space, discrete events occur. A Poisson is now recognized as a vitally important distribution in its own right.**

Letting p represents the likelihood of winning at given any attempt, the mean or the average number of wins in n tries will be given by π=np.

Using the binomial distribution of The Swiss mathematician Bernoulli, Poisson showed that the chance of winning k is about,

Formula:= p(k)=e^{–}^{π}.π^{k}/k!

Where e is the exponential function and k!=k(k-1)(k-2)…..1

**Condition to apply Poisson Distribution:-**

- The probability of more than one occurrence in the small interval is negligible (i.e. they are rare events).
- Every event must be at random and independent from others.
- The probability of the event taking place is proportional to the size of the interval for a small interval.
- Events are often failures, injuries, or extreme natural occurrences, such as Earthquakes where there is no theoretical upper limit of the number of incidents.

**Examples:-**

- The average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?
- The average number of computers sold by the Infotech Company is 15 homes per day. What is the probability that exactly 20 homes will be sold tomorrow?

**Main Differences Between** **Binomial Distribution and Poisson Distribution**

- The chances of success are constant in a binomial distribution, but there are very low chances of success in Poisson distribution.
- In the Binomial theorem, there is a fixed number of trials whereas in the Poisson theorem there is an infinite number of trials.
- Binomial Distribution is biparametric, i.e. it has two parameters n and p, while Poisson distribution is uniparametric, i.e. one parameter m.
- Each trial in binomial distribution is independent whereas in Poisson distribution the only number of occurrence in any given interval independent of others
- Binomial theorem used to predicts a number of successes within a set number of the trial at another side Poisson distribution it predicts the number of occurrences unit, time, space.

## Relationship Between the Binomial Distribution and the Poisson Distribution?

The binomial distribution tends toward the Poisson distribution as n → ∞, p → 0 and np stays constant.

If n is large and p is small, the Poisson distribution with λ= np closely approximates the binomial distribution.

Usually, the Poisson distribution is used to estimate the true underlying truth. Determining whether a random variable has a Poisson distribution can be difficult.

## Frequently Asked Questions (FAQ) About Binomial Distribution and Poisson Distribution

⚡ **What are the 4 properties of a binomial distribution?**

The 4 properties of a binomial distribution are:

- In a binomial distribution, trials are independent.
- There are only two outcomes for each and every trial in a binomial distribution. The two outcomes are success and failure.
- The probability of success in each trial is fixed.
- There is a fixed number of trials in a binomial distribution.

? **How do you identify a binomial distribution?**

In order to identify a binomial distribution, one must check if a variable has all the 4 properties of a binomial distribution or not. It means:

- There has to be a fixed number of trials.
- There could only be two outcomes either success or failure.
- The outcome of one trial does not affect the outcome of another hence, the trials are independent.
- The probability of success is the same for each trial.

? **Is rolling a die binomial distribution?**

Rolling a die could be a binomial distribution if the question related to dying roll is binary in nature.

For instance, if you ask that are you going to roll a 1 or not there will be just two outcomes for this.

Similarly, if you ask are you going to roll an even number or not, there will be just two outcomes for this.

Hence rolling a die is a binomial distribution in these instances.

But if you ask a question like if the number is going to be 1, 2 or 5 then it will not be a binomial distribution.

? **Is Poisson distribution continuous?**

Poisson distribution is not continuous. It is a discrete distribution. It becomes somewhat similar to a normal distribution if its mean is large.

? **Why is the Poisson distribution used?**

A Poisson distribution is used to statistically show how many times a rare event can occur in a given interval of time in a large population.

It can be used for the events that are occurring continuously within a given time period.

? **What is lambda in Poisson distribution?**

Lambda is a Poisson parameter that tells us an average number of events in a fixed time period. Its formula is:

λ = k/n

Here (k) is the number of events and (n) is the number of units.

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## Conclusion

After getting this much knowledge about these two distributions we can get a conclusion that apart from the aforementioned variations, these two distributions have a number of similar features, i.e. **both are the discrete theoretical distribution of probability.**

We can also use the Poisson distribution to find the waiting time between events. Even if we arrive at a random time, the average waiting time will always be the average time between events.

## Word Cloud for Difference Between Binomial Distribution and Poisson Distribution

The following is a collection of the most used terms in this article on the **Binomial Distribution and Poisson Distribution**. This should help in recalling related terms as used in this article at a later stage for you.

## References

- https://ieeexplore.ieee.org/abstract/document/930936
- https://arxiv.org/pdf/math/0511226.pdf
- https://projecteuclid.org/euclid.aoms/1177693172