# Statistics / Probability - Distribution - (Function)

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:

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.
• Head: the range of values where the pmf or pdf is relatively high.
• 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.
• Skewness: a measure of the extent to which a pmf or pdf “leans” to one side of its mean.

## Management

### Comparison

A Q-Q plot compare two distributions.

Example with ggplot current/stat_qq.html

``````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.

## Documentation / Reference

Discover More
(Probability|Statistics) - Binomial Distribution

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A histogram is a type of graph generally used to visualize a distribution An histogram is also known as a frequency distribution. Histograms can reveal information not captured by summary statistics...
Data Visualization - Box Plot

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Distribution - (Mean|Average) (M| | )

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A Measure of central tendency is a measure that describes the middle or center point of a distribution. A good measure of central tendency is representative of the distribution. The mean, the median and...
Distribution - Quantile Analysis

A quantile is a statistic that identifies the data that is less than the given value (ie that fall at or below a score in a distribution). A quantile function will always rank the data before giving any...
Frequency Distribution

A frequency distribution is a distribution of the frequency of each element ie the count of each element in a set or the count of each element in a period multiset
Mathematics - Probability distribution function

A Probability distribution function is a function that is used that specify relative likelihood (probability) of different outcomes of a single experiment. It assigns a probability (a nonnegative number)...
Monitoring Metrics - Distribution Summary

A distribution summary is a monitoring metrics type used to track the distribution of events. It is similar to a timer structurally, but records values that do not represent a unit of time. A distribution...