Maximal Entropy Distributions

When the mean of the distribution is known, and it is also known that the values are all positive, we can assume exponential distribution. 

University of Michigan - Ann Arbor

Data can be distributed in several ways. Some of these include:
- Normal (Gaussian) distributions
- Uniform distributions
- Exponential distributions

Data Distributions

Page, Scott. The Model Thinker. https://doi.org/10.1017/mag.2019.84

The Model Thinker

Bell-shaped distribution, often referred to as Gaussian. To test the assumption of normality, a variety of methods are available, but the simplest is to inspect the data visually using a histogram or a Q-Q scatterplot. Real-world data are almost never perfectly normal, so this assumption can be considered reasonably met if the shape looks approximately symmetric and bell-shaped. The data in the example figure below is approximately normally distributed.

Normal Distribution

https://machinelearningmastery.com/statistical-data-distributions/

A Gentle Introduction to Statistical Data Distributions

A distribution with constant probability.

Uniform Distribution

Exponential Distribution

Data Distributions: A Statistical Handbook (Entropy Minimax Sourcebook)
(https://www.amazon.com/Data-Distributions-Statistical-Handbook-Sourcebook/dp/0938876171)

Data Distributions: A Statistical Handbook (Entropy Minimax Sourcebook)

Distributions are often described by it's density functions.
There are two types of density functions:
1) Probability Density Function (PDF - Discrete)/Probability mass function(PMF-Continuous) 
2) Cumulattive Density Function (CDF)  

The graph below can shows the relationship between this two functions. For PDF, it will give you the probability one event happens. CDF gives you the probability an event equal or less than a value.


Density Function

A bimodal distribution is a specific shape of data distribution characterized by two distinct peaks, representing two scores or categories that occur most frequently. While less common than unimodal distributions in psychological research, they can emerge when different subsets of a sample respond divergently.

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