# Statistics / Probability - Distribution - (Function)

### Table of Contents

## About

This section talks about the term Distribution also knows as Probability distribution where you get:

- on the y axis, the probability
- on the x axis, the event

They can be seen as the outcomes of a single experiment.

The term “Probability'' asserts that each value in the set of possible values have different probabilities of being seen when reading/seeing a random variable.

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

In more technical terms, the probability distribution is a mathematical description of a random phenomenon (random variable?) in terms of the probabilities of events,

Many distributions are normal but not always. An histogram can help to find the type of distribution.

A box plot is a good summary of a distribution.

## Discrete / Continuous

### Discrete

There is two representation of a discrete distribution:

- the Bayesian representation: A discrete distribution plots just discrete values to probabilities such that the probabilities add up to 1.
- the frequentist representation. A infinite lists such that as n gets larger, sampling from the collection and counting the frequencies of each element approximates the Bayesian representation of the distribution.

### Continuous

standard continuous distributions— such as Gaussian, beta, binomial, and uniform.

algebraic properties, called conjugate priors. For example, a uniform prior combined with a binomial likelihood results in a beta posterior.

## Function

A distribution can be specified by supplying:

- a valid:
- probability mass function for continuous variable
- or probability density function for discrete variable

- a valid cumulative distribution function or survival function
- a valid hazard function
- a valid characteristic function
- a rule for constructing a new random variable from other random variables whose joint probability distribution is known.

Possible duplicate: Mathematics - Probability distribution function

## Characteristics

- Mode: for a discrete random variable, the value with highest probability (the location at which the probability mass function has its peak); for a continuous random variable, the location at which the probability density function has its peak.
- Support: the smallest closed set whose complement has probability zero.
- Tail: the complement of the head within the support; the large set of values where the pmf or pdf is relatively low.
- Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.
- Median: the value such that the set of values less than the median has a probability of one-half.
- Statistics - (Variance|Dispersion|Mean Square) (MS): the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.
- Standard deviation: the square root of the variance, and hence another measure of dispersion.
- Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value is a mirror image of the portion to its right.

## Type

- zipF/Pareto/Yule (Govern frequencies of different terms in a document, or web site visits)
- Gamma_distribution (Long tail → Latency distribution ?)
- Von_Mises_distribution around a circle.
- Weibull_distribution (to describe a particle size distribution, in reliability engineering and failure analysis)

## Management

### Comparison

A Q-Q plot compare two distributions.

Example with ggplot stat_qq

```
ggplot(res_succes, aes(sample=res_succes$TOTAL_TIME_SEC, colour = factor(res_succes$PRESENTATION_NAME))) +
geom_point(stat = "qq", size=0.75)
```

### Visualization

- A box plot is a good summary of a distribution.