# Probability Theory and Probability Distribution for Data Science and Analytics

Data science is the study of data for extracting meaningful insights for business. Data science and analytics have grown in popularity for getting insights and facts from datasets with methods, approaches, tools, and algorithms.

Businesses use this data for improving production, expanding business, and predicting customer needs.

Probability is a mathematical concept that predicts the likelihood of an event occurring. Understanding the probability theory and probability distribution is important for performing data analysis. This blog will discuss the concepts of probability theory and distribution in detail.

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## What is probability theory?

Probability theory is a branch of mathematics that studies the properties and behaviour of random phenomena, such as outcomes, events, distributions, and variables. Probability theory offers a framework for quantifying the likelihood of various scenarios, analysing the uncertainty and variability of data, and testing assumptions and hypotheses.

## Importance of probability theory in data analysis

Data is generally noisy, incomplete, or subject to errors and biases, making it difficult to draw reliable and accurate conclusions from it. Probability theory is necessary for data analysis as it helps in dealing with inherent variability and uncertainty of data.

With probability theory, it is easier to account for the sources of variability and uncertainty and to express confidence and certainty in the results. This theory also allows us to compare the different methods, models, and strategies for data analysis and for evaluating their validity and performance.

## Terms used in probability theory

In order to understand the application of probability theory, there are some terms that you must be familiar with. These are as follows:

### Random experiment

A random experiment can be defined as a trial that is repeated several times to get a well-defined set of possible outcomes. For example, tossing a coin.

### Sample space

It can be defined as the set of all possible outcomes that result from conducting a random experiment. For instance, the sample space of tossing a coin is (tails, head).

### Event

It can be defined as a set of outcomes of any particular experiment which forms a subset of the sample space. The different types of events are as follows:

• Independent events: The events which are not affected by any other events are called independent events.
• Dependent events: The events which are affected by other events are called independent events.
• Mutually exclusive events: The events which cannot take place at the same time are called mutually exclusive events.
• Equally likely events: Two or more events that have the same chance of taking place are called equally likely events.
• Exhaustive events: The event which is equal to the sample space of an experiment is called an exhaustive event.

### Random variable

A random variable, in probability theory, is a variable that considers the value of all possible results of an experiment. There are two kinds of random variables:

• Discrete random variable: These variables can be counted to an exact value like 0,1,2,...and so on.
• Continuous random variable: These variables can have an infinite number of values called the continuous random variable.

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## What are probability distributions?

It is a statistical function that defines all the possible values and probabilities of a random variable within a given range. This range is going to be bound by the minimum and maximum possible values. However, the possible values which are to be plotted on the probability distribution are going to be decided by several factors. Some of these factors are skewness, standard deviation, kurtosis, and average.

## Types of probability distributions

There are two kinds of probability distributions:

1. Discrete probability distributions
2. Continuous probability distributions

### Discrete probability distribution

This is a distribution where the observations can take only a finite number of values. For instance, the rolling of a dice can have only one number ranging from 1 to 6. There are several types of discrete distributions such as:

### Bernoulli distribution

In this type of distribution, only one experiment is conducted which results in a single observation. Hence, this type of distribution describes events that can have exactly two outcomes. For example, flipping a coin can have only one of the two outcomes - heads or tails.

### Binomial distribution

In this type of distribution, there can be a finite number of possibilities. It is like an extended version of Bernoulli’s distribution. Repeating the Bernoulli trials, n number of times, we will get a binomial distribution.

### Poisson distribution

This is a type of distribution used in statistics to show how many times an event is likely to occur over a given period. Poisson distributions are generally used for comprehending independent events at a constant rate during defined time intervals.

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### Continuous probability distributions

This type of distribution can define the probabilities of the possible values of a continuous random variable. Continuous distributions have smooth curves, unlike discrete distributions, which have an infinite number of samples.

### Normal distribution

Also known as the Gaussian distribution, this is the most common and naturally occurring distribution. This distribution is seen in almost every field - statistics, finance, chemistry, etc. This probability distribution is symmetrical around its mean (average) value. It also signifies that the data close to the mean occurs more frequently than the data that is far from it.

### Exponential distribution

An exponential distribution, in a Poisson process, is a continuous probability distribution that describes the time period between the events occurring.

### Continuous uniform distribution

In this type of distribution, all the outcomes are equally possible. Every variable has the chance of occurring as a result. In this symmetric distribution, the variables are spaced evenly, having a 1/(b-a) probability.

### Log-normal distribution

This is a continuous distribution of random variables, whereas the natural logarithms of these random variables are a normal distribution. A log-normal distribution is always going to yield a positive value as opposed to a normal distribution.

#### Conclusion

Probability is an estimation of how likely an event or outcome can occur. Probability theory serves as the backbone of a number of data science concepts. Probability theory deals with the uncertainty associated with data.

The probability distribution is the set of all the possible outcomes of any random event or experiment. It has many real-life applications in areas such as engineering, business, medicine, and many more industries. It is used mainly to make future predictions based on a sample for a random event.

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