In statistics, economics and many other areas, it is necessary to infer and decide on situations in which there are different probabilities of occurrence in the results, the **probability distribution** allows, from a function, to describe the expected behavior in those cases. Different types of probability distribution

## What is the probability distribution?

The probability distribution refers to all the possible outcomes that a random variable may have, that is, it describes the behavior of said variable within a range of values or possible outcomes.

The random variable can be discrete or continuous. A discrete random variable is one represented by integers, characterized by the limit of values it can take. On the other hand, a continuous random variable does not have this separation or limitation, it can take any value within the established limit.

## Importance of the Study of the Distribution of Probabilities

The probability distribution allows assigning to each event the probability that it occurs or is successful, an example of this, conducting experiments, studies on the progress of a company, etc.

With the study of probabilities, a way has been allowed to standardize the events and processes that occur at random, this has been achieved by estimating the frequencies in which a specific result is obtained. Different types of probability distribution

## Distribution Types

The type of distribution depends on the type of variable that is being treated. There are many, below, the main or best known:

- For continuous variables: in the case that the random variable is continuous, the associated distribution is a normal or Gaussian type distribution.
- For discrete variables: in the case that the random variable is discrete, there may be several types of distributions, the main ones being the binomial distribution, the hypergeometric distribution and the Poisson distribution. Different types of probability distribution

### Normal distribution

It is one of the most important in the area of statistics. Its development and explanation are attributed to different researchers, especially Carl Friedrich Gauss.

This distribution considers two parameters, which are the mean or mean ( *μ* ) and the standard deviation *(σ* ). Thanks to these two parameters, it has an associated equation, from which a graph known as a Gaussian bell is developed.

This graph is symmetric with respect to the mean and its opening or width is given by the standard deviation. In turn, the graph reflects **the probability distribution** of the variable under study.

From this normal distribution three other types of distributions are developed:

- Student’s t
- Chi-square
- Fisher’s F

#### Examples of Normal Distribution

Some examples where a normal distribution can occur are:

- The effect of a medicine or drug. Different types of probability distribution
- The change in temperature at a specific time of year.
- Morphological characters such as weight or height in a group of individuals.

### Binomial Distribution

It was developed by Jacob Bernoulli, it has various applications in the area of biostatistics, specifically in conducting experiments, it is also known as the Bernoulli distribution.

An experiment or study has a binomial distribution when the following conditions are met:

- In the experiment there are only two possible outcomes, success or failure.
- Repeating the same experiment presents a result that is independent of the previous results.
- The probability of success or failure is constant.
- Each experiment has the same number of replications. Different types of probability distribution

#### Examples of Binomial Distribution

It is applied to experiments and relationships in the areas of medicine or biology, although it can also be applied in finance and economics. Some examples of its application are:

- Whether or not a person has a disease such as cancer, smallpox, or hepatitis.
- Whether or not a woman is pregnant.
- Whether the publication of an article was successful or not.

### Hypergeometric Distribution

This type of distribution is related to non-replacement and random sampling. In sampling without replacement, no selected item is returned or discarded until the sampling is complete.

In turn, this type of distribution occurs in cases where the absence or presence of some characteristic is investigated. Different types of probability distribution

It is similar to the binomial, but in the case of the hypergeometric, the probability associated with each result does not remain constant, this due to the characteristic of sampling without replacement. However, if the number of samples is very large, the distribution can be close to a binomial.

#### Hypergeometric Distribution Examples

It is common to have this type of distribution in relatively small population samples. Some examples where a distribution of this type occurs can be:

- When the quality control of a company is carried out, which can depend and vary according to the manufacturer.
- The control of defective instruments in an office or company.
- When you want to know the probability of choosing a defective instrument or object.

### Poisson distribution

It was developed by Siméon Denis Poisson, this type of distribution explains the probability that a certain event occurs a certain number of times in a set time.

In general, this type of distribution occurs when the appearance of some rare event or event is observed in said established time. Different types of probability distribution

In addition to being viewed as the probability at a set time, it can also be viewed as the probability of success in a unit area or product number.

In this type of distribution, the probability of success is also independent in each established interval, so it is not constant. Some event or process involving a Poisson distribution is stable.

On the other hand, knowing the number of events that occur in a set interval does not mean that you can predict the number of events that will occur in the next.

#### Poisson Distribution Examples

This type of distribution is observed in different processes, some examples of this can be:

- When it is desired to study the probability or the number of times that an adverse reaction to the application of a drug may occur.
- When it is required to know the number of defects in a batch of fabric. In this case, the established interval would be a unit of area.
- When it is intended to know the number of bacteria per unit area in a culture.
- When the number of boat arrivals at a particular site is expected to be known.

Finally, it is worth highlighting the great help that this statistical analysis, development and technological advances have meant, since they take a lot of work if done by hand, but there are programs and applications capable of generating the necessary information to be able to interpret and respond to the problem raised. Different types of probability distribution