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# Probability Distribution

## Introduction

Probability distribution is a statistical function that describes the likelihood of obtaining different outcomes from a random experiment or process. It specifies the possible values that a random variable can take and the probability of each value occurring.

## What is Probability Distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes in a random experiment or event. Probability distributions can be either discrete or continuous. A discrete probability distribution describes the probability of each possible outcome of a discrete random variable, which takes on a finite or countably infinite set of possible values.

For example, the probability distribution of the outcome of rolling a fair six-sided die is a discrete uniform distribution, where each outcome has an equal probability of 1/6.

## Types of Probability Distribution

There are two types of probability distributions: discrete and continuous.

## 1. Discrete Probability Distribution

A discrete probability distribution is one in which the possible values of the random variable are countable and finite or infinite. Examples of discrete probability distributions include the binomial distribution, Poisson distribution, and geometric distribution.

• Binomial Distribution: The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials. The probability of success in each trial is denoted by p, and the probability of failure is denoted by q = 1 – p. The binomial distribution has two parameters: n, the number of trials, and p, the probability of success.
• Poisson Distribution: The Poisson distribution is a discrete probability distribution that describes the number of occurrences of an event in a fixed interval of time or space. The Poisson distribution has one parameter: λ, the average rate of occurrence.
• Geometric Distribution: The geometric distribution is a discrete probability distribution that describes the number of trials needed to obtain the first success in a series of independent trials. The probability of success in each trial is denoted by p, and the probability of failure is denoted by q = 1 – p. The geometric distribution has one parameter: p, the probability of success.

## 2. Continuous Probability Distribution

A continuous probability distribution is one in which the possible values of the random variable are uncountable and infinite. Examples of continuous probability distributions include the normal distribution, exponential distribution, and uniform distribution.

• Normal Distribution: The normal distribution is commonly used in statistics to model many natural phenomena. The normal distribution is characterized by two parameters: μ, the mean, and σ, the standard deviation.
• Exponential Distribution: The exponential distribution is a continuous probability distribution that describes the time between occurrences of a random event. The exponential distribution has one parameter: λ, the rate parameter.
• Uniform Distribution: The uniform distribution is a continuous probability distribution in which all values within a given range are equally likely to occur. The uniform distribution has two parameters: a, the lower bound, and b, the upper bound.