What is Probability Distribution? How Does One Use it in Real Life?

A fundamental concept in statistics and data analysis, probability plays a crucial role in understanding and predicting the outcome of random events. Apart from taking a closer look at what is probability distribution, this guide also delves into the basics of probability distribution and explores its properties and characteristics. We will also look at different types of probability distributions—Bernoulli, Binomial, Poisson, Normal, etc.—and see how they are used to model different data types.

What is Probability Distribution?

This function describes the probability of each possible value of a random variable. It assigns a probability to each possible outcome of a random event and provides a method for summarizing and analyzing random variable behavior. Probability distribution, has two functions, Probability Density Function (PDF) and a Cumulative Distribution Function (CDF), they are used to predict the values a random variable might take in a given experiment. Different probability distributions, such as the Bernoulli, Binomial, Poisson, and Normal, are used to model various data types, and these have different properties and characteristics.

What is Probability Distribution Used for?

Probability distribution models and evaluates numerous real-world phenomena and anticipates future outcomes based on previous observations and trends. Statistics, economics, finance, engineering, and natural sciences all employ probability distributions to understand and make decisions when clouded by uncertainty. For example, predicting an outcome when a coin is flipped.

Types of Probability Distribution

Now that we know what is probability distribution, let’s study its two main types:

Discrete Distributions

This model is used to count data and can take only a limited number of possible values. Examples of discrete distributions are:

• Bernoulli Distribution
• Binomial Distribution
• Poisson Distribution

Continuous Distributions

They are used to model data that can take any value within a range. Examples of continuous distributions are:

• Uniform Distribution
• Normal (Gaussian) Distribution
• Exponential Distribution

The Formula for Probability Distribution

The probability distribution formula depends on the type of distribution it is. Here are a few examples of common probability distributions and their formulae:

Bernoulli Distribution

Formula

f(k;p) =pk + (1-p)(1-k)

p = probability

k = possible outcomes

f = probability mass function

Binomial Distribution:

P_{x} = {n \choose x} p^{x} q^{n-x}

P = binomial probability

x = number of times for a specific outcome within n trials

{n \choose x} = number of combinations

p = probability of success on a single trial

q = probability of failure on a single trial

n = number of trials

Poisson Distribution

P(x=k) = (e^(-λ )) * (λ ^k) / k!

Here, λ is the average rate of success, and “k” is the number of successes.

Normal (Gaussian) Distribution

P(x) = (1 / (sqrt(2 * pi * sigma^2))) * e^-((x – mu)^2 / (2 * sigma^2))

Here, “mu” is the mean, “sigma” is the standard deviation, and “x” is a continuous random variable.

Characteristics of Probability Distribution

A probability distribution is a function that describes the likelihood of obtaining various values for a random variable. Depending on whether the random variable can take specific, discrete values or a range of values, it can be discrete or continuous. A probability distribution’s probabilities must add up to one, and each random variable value must have a non-negative probability. The parameters that describe the distribution’s underlying characteristics determine its shape. Furthermore, probability distributions can be used to compute statistics like the mean, variance, and quantiles, which provide additional insight into the distribution’s behavior.

Probability Distribution Examples and Solutions

Here are a few examples of common probability distributions with their solutions:

Bernoulli Distribution

A coin flip is an example of a Bernoulli experiment, with heads indicating success and tails indicating failure.

The chance of getting heads is 0.5, as is the chance of getting tails. Moreover, since the mean of a Bernoulli distribution equals its success probability, the expected value is 0.5.

Binomial Distribution

Consider the following experiment: A person flips a coin 10 times and counts the number of heads. Essentially, this is because the mean of a Binomial distribution is equal to n * p, where n is the number of trials and p is the probability of success in a single trial, the expected value of this distribution is five heads.

This distribution’s variance equals n * p * (1 – p).

Poisson Distribution

The number of calls received by a call center: A call center receives an average of 10 calls per hour. In fact, sing the Poisson distribution, we can calculate the probability of receiving 0, 1, 2, 3, or more calls in a given hour. For example, the following explains the probability of receiving exactly 12 calls in an hour:

P(X = 12) = (e^-10 * 10^12) / 12! = 0.067

Properties of Probability Distribution Function

The probability density function’s properties are:

• The function should be greater than 0
• The total area under the function’s curve is equal to 1
• Any real positive number can be the function

In other words, a probability density function cannot be f ( x ) = – 2 or f ( x ) = i. (where i = imaginary number)

Probability distributions play a crucial role in understanding and analyzing the behavior of random variables. We hope this guide has shed light on what is probability distribution and its types. To learn more about probability distribution, check out these online data science and analytics courses Emeritus offers in tie-up with some of the best universities worldwide.

By Siddhesh Shinde

Write to us at content@emeritus.org

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