# Probability Distributions

**Probability distribution**:- It is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. There are many types of Probability distribution,such as:-

Bernoulli Distribution

Binomial Distribution

Multivariate Normal Distributions

Poisson Distribution

Exponential Distribution

Beta Distribution

**Bernoulli Distribution**:- It is a discrete distribution having two possible outcomes labeled by n=0 and n=1 in which n=1 ("success") occurs with probability p and n=0 ("failure") occurs with probability q=1-p, where 0<p<1. The outcome is either 0 or 1 always.

Probability mass function of Bernoulli distribution is :- P(n)=p^n*(1−p)^(1−n)

**Binomial Distribution**:- It is defined as when there is only one outcome for each trail, that each trail has the same probability of success, and that each trial is mutually exclusive or independent of the other.

Its probability distribution function is given by :

k= number of events ( if interest)

n= total number of events

p= probability of success of event

Example of binomial distribution-

**Question- **There are 10 sets of traffic lights on the journey. The probability that a driver must stop at any one traffic light coming to airport is 0.25

What is the probability that a driver must stop at exactly 2 of the 10 sets of traffic lights?

**Answer- **

**Poisson Distribution**:- The Poisson distribution is used to model the number of events occurring within a given time interval. Events are happening at constant rate and independent of each other( like road accidents, patients coming at hospital etc.) .The formula for the Poisson probability mass function( it's *discrete probability distribution)* is.

Example of Poisson's distribution-

**Question- **The number of work related injuries per month in a manufacturing plant is known to follow a Poisson distribution, with a mean of 2.5 work-related injuries a month. What is the probability that in a given month, no work-related injuries occur?

**Answer- **

**Exponential Distribution- **It is the probability distribution of the time between events in Poisson's distribution. A continuous random variable X is said to have an exponential distribution with parameter λ>0, shown as X∼Exponential(λ), if its PDF is given by

**Normal Distribution- **The normal distribution , also known as the Gaussian distribution, is so called because its based on the Gaussian function . This distribution is defined by two parameters: the mean( mu) , which is the expected value of the distribution, and the standard deviation(sigma) , which corresponds to the expected deviation from the mean.

Probability Density Function is given by-

Above function is for single variable, normal distribution may have multiple random variables that may be even co-related.

**Beta Distribution-** It is defined on the interval [0,1] denoted by α and β, usually. α and β are two positive parameters that appear as exponents of the random variable and is intended to control the shape of the distribution. Its notation is Beta(α,β), where α and β are the real numbers, and the values are more than zero. Probability density function for various values of α and β-

PDF is given by-