Question

What is the difference between binomial distribution and Poisson distribution?

Answer 1

Hint : Binomial distribution is the one in which the number of outcomes are only 2 and whereas Poisson distribution is the one in which the number of outcomes are unlimited. The difference, definition and examples of both the distributions are discussed below.

Complete step by step solution:
In this question, we are going to see the defination, comparison, differences and some examples of binomial distribution and Poisson distribution.
First of all, let us see the definition of binomial distribution and Poisson distribution with examples.

Binomial Distribution:
Binomial distribution is the one in which the number of outcomes are only two, that is success or failure.
It is widely used and derived from the Bernoulli process. Now, binomial distribution has two parameters n and p, where n means number of trials and p means the success probability. Now, the mean and variance of binomial distribution are denoted by
$\Rightarrow {\mu _x} = np$
$\Rightarrow {\sigma ^2}_x = npq$
Example of binomial distribution: Coin toss.

Poisson distribution:
Poisson distribution is the one in which the number of possible outcomes has no limits.
It describes the probability of certain events occurring in some time interval. It is uniparametric distribution as it has only one parameter $\lambda$ or m. The mean and the variance of the poisson distribution are given by
$\Rightarrow {\mu _x} = m$ or $\lambda$
$\Rightarrow {\sigma ^2}_x = m$ or $\lambda$
Poisson distribution is used when the number of events is high and the probability of its occurrence is quite low.
Example of Poisson distribution: Number of insurance claims per day on an insurance company.

Now, let us see the key differences between binomial distribution and Poisson distribution.

Binomial Distribution	Poisson Distribution
It is biparametric, i.e. it has 2 parameters n and p.	It is uniparametric, i.e. it has only 1 parameter m.
The number of attempts are fixed.	The number of attempts are unlimited.
The probability of success is constant.	The probability of success is extremely small.
There are only two possible outcomes-Success or failure.	There are unlimited possible outcomes.
Mean>Variance	Mean=Variance

Note : The probability of binomial distribution is represented by
$\Rightarrow P\left( {X = x} \right) = {}^n{C_x}{p^x}{q^{n - x}},x = 0,1,2,3,...n$
The probability of the Poisson distribution is represented by
$\Rightarrow P\left( {X = x} \right) = \dfrac{{{e^{ - m}} \cdot {m^x}}}{{x!}},x = 0,1,2,3,...$
Here, $e = 2.718$