A *probability distribution* is a function that describes the chances of finding a random variable over a defined range.

Below is the list of some of the most used probability distribution functions.

• Normal(Gaussian) distribution.

• Log-Normal distribution.

• Bernoulli distribution.

• Binomial distribution.

• Poisson distribution.

Today let’s see what are Bernoulli, Binomial and Poisson distribution.

### Bernoulli distribution:-

The distribution of random variables which can take up two values, i.e. either success(1) or failure(0) is called a Bernoulli distribution. Some examples are coin flip, whether a patient tested has cancer or not, etc.

Consider we have a random variable X, which follows Bernoulli distribution. It can be denoted as; **X~Ber(p)**

The probabilities are given as:-

For success: P(X=1)=p

For failure: P(X=0)=1-p

The standard deviation is given by; Standard Deviation=√[p(1-p)]

Variance is the square of standard deviation and hence Variance=p(1-p)

The expectation for Bernoulli distribution is given as E[X]=p

### Binomial distribution:-

A binomial distribution is a distribution of random variables that represent the number of success or failures in 'n' successive independent trials of a Bernoulli experiment. Examples include coin flips for a particular ’n’ number of times, result whether 'n' number of patients have hypertension or not.

Thus we can say that binomial distribution is some complex version of Bernoulli distribution.

Suppose X is a random variable following Binomial distribution. It can be denoted as- **X~Bin(n,p)**

Probability is given as below-

Expectation E[x]=np

Variance is given by var(X)=np(1-p)

### Poisson distribution:-

The Poisson distribution is the special case of binomial distribution where p → 0 and n → ∞.

It is used when we have to express the probability of a given set of event occurs in a fixed time interval or a constant space with a fixed mean rate. In addition to this, the event happening should be independent of the time since the occurrence of the last event.

The probability function is given as;

Expectation E[X]=λ where np → λ as p →0 and n →∞.

Here Standard Deviation=√λ

## Discussion