Probability distribution

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https://en.wikipedia.org/wiki/Probability_distribution

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values.

Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names.

Introduction

A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by ${\displaystyle \mathrm{\Omega }}$, is the set of all possible outcomes of a random phenomenon being observed; it may be any set: a set of real numbers, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip would be Ω = {heads, tails}.

To define probability distributions for the specific case of random variables (so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous random variables. In the discrete case, it is sufficient to specify a probability mass function ${\displaystyle p}$ assigning a probability to each possible outcome: for example, when throwing a fair dice, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is

$${\displaystyle p(2)+p(4)+p(6)=1/6+1/6+1/6=1/2.}$$

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale has many digits of precision. The probability that it weighs exactly 500 g is zero, as it will most likely have some non-zero decimal digits. Nevertheless, one might demand, in quality control, that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability, and this demand is less sensitive to the accuracy of measurement instruments.

The left graph shows a probability density function. The right graph shows the cumulative distribution function, for which the value at a equals the area under the probability density curve to the left of a.

Absolutely continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. An alternative description of the distribution is by means of the cumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e., ${\displaystyle P(X<x)}$ for some ${\displaystyle x}$). The cumulative distribution function is the area under the probability density function from ${\displaystyle -\mathrm{\infty }}$ to ${\displaystyle x}$, as described by the picture to the right.

General probability definition

A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function ${\displaystyle P:\mathcal{A}\to \mathbb{R}}$ whose input space ${\displaystyle \mathcal{A}}$ is a σ-algebra, and gives a real number probability as its output, particularly, a number in ${\displaystyle [0,1]\subseteq \mathbb{R}}$.

The probability function ${\displaystyle P}$ can take as argument subsets of the sample space itself, as in the coin toss example, where the function ${\displaystyle P}$ was defined so that P(heads) = 0.5 and P(tails) = 0.5. However, because of the widespread use of random variables, which transform the sample space into a set of numbers (e.g., ${\displaystyle \mathbb{R}}$, ${\displaystyle \mathbb{N}}$), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It is common to denote as ${\displaystyle P(X\in E)}$ the probability that a certain value of the variable ${\displaystyle X}$ belongs to a certain event ${\displaystyle E}$.

The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms, that is:

1. ${\displaystyle P(X\in E)\ge 0\mathrm{\forall }E\in \mathcal{A}}$, so the probability is non-negative
2. ${\displaystyle P(X\in E)\le 1\mathrm{\forall }E\in \mathcal{A}}$, so no probability exceeds ${\displaystyle 1}$
3. ${\displaystyle P(X\in \bigcup _i{E}_i)=\sum _{i}P(X\in E_i)}$ for any countable disjoint family of sets ${\displaystyle \{E_i\}}$

The concept of probability function is made more rigorous by defining it as the element of a probability space ${\displaystyle (X,\mathcal{A},P)}$, where ${\displaystyle X}$ is the set of possible outcomes, ${\displaystyle \mathcal{A}}$ is the set of all subsets ${\displaystyle E\subset X}$ whose probability can be measured, and ${\displaystyle P}$ is the probability function, or probability measure, that assigns a probability to each of these measurable subsets ${\displaystyle E\in \mathcal{A}}$.

Probability distributions usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function. On the other hand, absolutely continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In the absolutely continuous case, probabilities are described by a probability density function, and the probability distribution is by definition the integral of the probability density function. The normal distribution is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various different values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. A commonly encountered multivariate distribution is the multivariate normal distribution.

Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.

The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important absolutely continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.

Terminology

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.

Basic terms

• Random variable: takes values from a sample space; probabilities describe which values and set of values are taken more likely.
• Event: set of possible values (outcomes) of a random variable that occurs with a certain probability.
• Probability function or probability measure: describes the probability ${\displaystyle P(X\in E)}$ that the event ${\displaystyle E,}$ occurs.
• Cumulative distribution function: function evaluating the probability that ${\displaystyle X}$ will take a value less than or equal to ${\displaystyle x}$ for a random variable (only for real-valued random variables).
• Quantile function: the inverse of the cumulative distribution function. Gives ${\displaystyle x}$ such that, with probability ${\displaystyle q}$, ${\displaystyle X}$ will not exceed ${\displaystyle x}$.

Discrete probability distributions

• Discrete probability distribution: for many random variables with finitely or countably infinitely many values.
• Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value.
• Frequency distribution: a table that displays the frequency of various outcomes in a sample.
• Relative frequency distribution: a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size).
• Categorical distribution: for discrete random variables with a finite set of values.

Absolutely continuous probability distributions

• Absolutely continuous probability distribution: for many random variables with uncountably many values.
• Probability density function (pdf) or probability density: function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

Related terms

• Support: set of values that can be assumed with non-zero probability by the random variable. For a random variable ${\displaystyle X}$, it is sometimes denoted as ${\displaystyle R_X}$.
• Tail: the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form ${\displaystyle X>a}$, ${\displaystyle X<b}$ or a union thereof.
• Head: the region where the pmf or pdf is relatively high. Usually has the form ${\displaystyle a<X<b}$.
• 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, and the set greater than the median, each have probabilities no greater than one-half.
• Mode: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
• Quantile: the q-quantile is the value ${\displaystyle x}$ such that ${\displaystyle P(X<x)=q}$.
• Variance: 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 (usually the median) 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. The third standardized moment of the distribution.
• Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

Cumulative distribution function

In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable ${\displaystyle X}$ with regard to a probability distribution ${\displaystyle p}$ is defined as

$${\displaystyle F(x)=P(X\le x).}$$

The cumulative distribution function of any real-valued random variable has the properties:

• ${\displaystyle F(x)}$ is non-decreasing;
• ${\displaystyle F(x)}$ is right-continuous;
• ${\displaystyle 0\le F(x)\le 1}$;
• ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}F(x)=0}$ and ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}F(x)=1}$; and
• ${\displaystyle Pr(a<X\le b)=F(b)-F(a)}$.

Conversely, any function ${\displaystyle F:\mathbb{R}\to \mathbb{R}}$ that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.

Any probability distribution can be decomposed as the mixture of a discrete, an absolutely continuous and a singular continuous distribution, and thus any cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions.

Discrete probability distribution

Main article: Probability mass function

The probability mass function (pmf) ${\displaystyle p(S)}$ specifies the probability distribution for the sum ${\displaystyle S}$ of counts from two dice. For example, the figure shows that ${\displaystyle p(11)=2/36=1/18}$. The pmf allows the computation of probabilities of events such as ${\displaystyle P(X>9)=1/12+1/18+1/36=1/6}$, and all other probabilities in the distribution.

The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.

The cdf of a discrete probability distribution, ...

... of a continuous probability distribution, ...

... of a distribution which has both a continuous part and a discrete part

A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values (almost surely) which means that the probability of any event ${\displaystyle E}$ can be expressed as a (finite or countably infinite) sum:

$${\displaystyle P(X\in E)=\sum _{\omega \in A\cap E}P(X=\omega ),}$$

where ${\displaystyle A}$ is a countable set with ${\displaystyle P(X\in A)=1}$. Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a probability mass function ${\displaystyle p(x)=P(X=x)}$. In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if ${\displaystyle p(n)=\frac{1}{2^n}}$ for ${\displaystyle n=1,2,...}$, the sum of probabilities would be ${\displaystyle 1/2+1/4+1/8+\cdots =1}$.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution. When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

Cumulative distribution function

A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take. Thus the cumulative distribution function has the form

$${\displaystyle F(x)=P(X\le x)=\sum _{\omega \le x}p(\omega ).}$$

The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.

Dirac delta representation

A discrete probability distribution is often represented with Dirac measures, the probability distributions of deterministic random variables. For any outcome ${\displaystyle \omega }$, let ${\displaystyle {\delta }_{\omega }}$ be the Dirac measure concentrated at ${\displaystyle \omega }$. Given a discrete probability distribution, there is a countable set ${\displaystyle A}$ with ${\displaystyle P(X\in A)=1}$ and a probability mass function ${\displaystyle p}$. If ${\displaystyle E}$ is any event, then

$${\displaystyle P(X\in E)=\sum _{\omega \in A}p(\omega ){\delta }_{\omega }(E),}$$

or in short,

$${\displaystyle P_X=\sum _{\omega \in A}p(\omega ){\delta }_{\omega }.}$$

Similarly, discrete distributions can be represented with the Dirac delta function as a generalized probability density function ${\displaystyle f}$, where

$${\displaystyle f(x)=\sum _{\omega \in A}p(\omega )\delta (x-\omega ),}$$

which means

$${\displaystyle P(X\in E)={\int }_{E}f(x)dx=\sum _{\omega \in A}p(\omega ){\int }_E\delta (x-\omega )=\sum _{\omega \in A\cap E}p(\omega )}$$

for any event ${\displaystyle E.}$

Indicator-function representation

For a discrete random variable ${\displaystyle X}$, let ${\displaystyle u_0,u_1,\dots }$ be the values it can take with non-zero probability. Denote

$${\displaystyle {\mathrm{\Omega }}_i=X^{-1}(u_i)=\{\omega :X(\omega )=u_i\},i=0,1,2,\dots }$$

These are disjoint sets, and for such sets

$${\displaystyle P\left(\bigcup _i{\mathrm{\Omega }}_i\right)=\sum _{i}P({\mathrm{\Omega }}_i)=\sum _{i}P(X=u_i)=1.}$$

It follows that the probability that ${\displaystyle X}$ takes any value except for ${\displaystyle u_0,u_1,\dots }$ is zero, and thus one can write ${\displaystyle X}$ as

$${\displaystyle X(\omega )=\sum _i{u}_i{1}_{{\mathrm{\Omega }}_i}(\omega )}$$

except on a set of probability zero, where ${\displaystyle 1_A}$ is the indicator function of ${\displaystyle A}$. This may serve as an alternative definition of discrete random variables.

One-point distribution

A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a deterministic distribution. Expressed formally, the random variable ${\displaystyle X}$ has a one-point distribution if it has a possible outcome ${\displaystyle x}$ such that ${\displaystyle P(X=x)=1.}$ All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.

Absolutely continuous probability distribution

Main article: Probability density function

An absolutely continuous probability distribution is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. More precisely, a real random variable ${\displaystyle X}$ has an absolutely continuous probability distribution if there is a function ${\displaystyle f:\mathbb{R}\to [0,\mathrm{\infty }]}$ such that for each interval ${\displaystyle [a,b]\subset \mathbb{R}}$ the probability of ${\displaystyle X}$ belonging to ${\displaystyle [a,b]}$ is given by the integral of ${\displaystyle f}$ over ${\displaystyle I}$:

$${\displaystyle P\left(a\le X\le b\right)={\int }_a^{b}f(x)dx.}$$

This is the definition of a probability density function, so that absolutely continuous probability distributions are exactly those with a probability density function. In particular, the probability for ${\displaystyle X}$ to take any single value ${\displaystyle a}$ (that is, ${\displaystyle a\le X\le a}$) is zero, because an integral with coinciding upper and lower limits is always equal to zero. If the interval ${\displaystyle [a,b]}$ is replaced by any measurable set ${\displaystyle A}$, the according equality still holds:

$${\displaystyle P(X\in A)={\int }_{A}f(x)dx.}$$

An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous.

There are many examples of absolutely continuous probability distributions: normal, uniform, chi-squared, and others.

Cumulative distribution function

Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution function. In this case, the cumulative distribution function ${\displaystyle F}$ has the form

$${\displaystyle F(x)=P(X\le x)={\int }_{-\mathrm{\infty }}^{x}f(t)dt}$$

where ${\displaystyle f}$ is a density of the random variable ${\displaystyle X}$ with regard to the distribution ${\displaystyle P}$.

Note on terminology: Absolutely continuous distributions ought to be distinguished from continuous distributions, which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but the inverse is not true, there exist singular distributions, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the Cantor distribution. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions.

For a more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure.

Kolmogorov definition

Main articles: Probability space and Probability measure

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function ${\displaystyle X}$ from a probability space ${\displaystyle (\mathrm{\Omega },\mathcal{F},\mathbb{P})}$ to a measurable space ${\displaystyle (\mathcal{X},\mathcal{A})}$. Given that probabilities of events of the form ${\displaystyle \{\omega \in \mathrm{\Omega }\mid X(\omega )\in A\}}$ satisfy Kolmogorov's probability axioms, the probability distribution of ${\displaystyle X}$ is the image measure ${\displaystyle X_{\ast }\mathbb{P}}$ of ${\displaystyle X}$ , which is a probability measure on ${\displaystyle (\mathcal{X},\mathcal{A})}$ satisfying ${\displaystyle X_{\ast }\mathbb{P}=\mathbb{P}{X}^{-1}}$.

Other kinds of distributions

One solution for the Rabinovich–Fabrikant equations. What is the probability of observing a state on a certain place of the support (i.e., the red subset)?

Absolutely continuous and discrete distributions with support on ${\displaystyle {\mathbb{R}}^k}$ or ${\displaystyle {\mathbb{N}}^k}$ are extremely useful to model a myriad of phenomena, since most practical distributions are supported on relatively simple subsets, such as hypercubes or balls. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves ${\displaystyle \gamma :[a,b]\to {\mathbb{R}}^n}$ within some space ${\displaystyle {\mathbb{R}}^n}$ or similar. In these cases, the probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it.

One example is shown in the figure to the right, which displays the evolution of a system of differential equations (commonly known as the Rabinovich–Fabrikant equations) that can be used to model the behaviour of Langmuir waves in plasma. When this phenomenon is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system.

This kind of complicated support appears quite frequently in dynamical systems. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let ${\displaystyle t_1\ll t_2\ll t_3}$ be instants in time and ${\displaystyle O}$ a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set ${\displaystyle O}$ would be equal in interval ${\displaystyle [t_1,t_2]}$ and ${\displaystyle [t_2,t_3]}$, which might not happen; for example, it could oscillate similar to a sine, ${\displaystyle \sin (t)}$, whose limit when ${\displaystyle t\to \mathrm{\infty }}$ does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. The branch of dynamical systems that studies the existence of a probability measure is ergodic theory.

Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely continuous" or "discrete" depending on whether the support is uncountable or countable, respectively.

Random number generation

Main article: Pseudo-random number sampling

Most algorithms are based on a pseudorandom number generator that produces numbers ${\displaystyle X}$ that are uniformly distributed in the half-open interval [0, 1). These random variates ${\displaystyle X}$ are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.

For example, suppose ${\displaystyle U}$ has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some ${\displaystyle 0<p<1}$, we define

$${\displaystyle X=\{\begin{array}{ll}1,& \text{if }U<p\\ 0,& \text{if }U\ge p\end{array}}$$

so that

$${\displaystyle Pr(X=1)=Pr(U<p)=p,Pr(X=0)=Pr(U\ge p)=1-p.}$$

This random variable X has a Bernoulli distribution with parameter ${\displaystyle p}$. This is a transformation of discrete random variable.

For a distribution function ${\displaystyle F}$ of an absolutely continuous random variable, an absolutely continuous random variable must be constructed. ${\displaystyle F^{\mathit{i}\mathit{n}\mathit{v}}}$, an inverse function of ${\displaystyle F}$, relates to the uniform variable ${\displaystyle U}$:

$${\displaystyle U\le F(x)=F^{\mathit{i}\mathit{n}\mathit{v}}(U)\le x.}$$

For example, suppose a random variable that has an exponential distribution ${\displaystyle F(x)=1-e^{-\lambda x}}$ must be constructed.

$${\displaystyle \begin{array}{rl}F(x)=u& \iff 1-e^{-\lambda x}=u\\ & \iff e^{-\lambda x}=1-u\\ & \iff -\lambda x=\ln (1-u)\\ & \iff x=\frac{-1}{\lambda }\ln (1-u)\end{array}}$$

so ${\displaystyle F^{\mathit{i}\mathit{n}\mathit{v}}(u)=\frac{-1}{\lambda }\ln (1-u)}$ and if ${\displaystyle U}$ has a ${\displaystyle U(0,1)}$ distribution, then the random variable ${\displaystyle X}$ is defined by ${\displaystyle X=F^{\mathit{i}\mathit{n}\mathit{v}}(U)=\frac{-1}{\lambda }\ln (1-U)}$. This has an exponential distribution of ${\displaystyle \lambda }$.

A frequent problem in statistical simulations (the Monte Carlo method) is the generation of pseudo-random numbers that are distributed in a given way.

Common probability distributions and their applications

For a more comprehensive list, see List of probability distributions.

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, absolutely continuous, multivariate, etc.)

All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution.

Linear growth (e.g. errors, offsets)

• Normal distribution (Gaussian distribution), for a single such quantity; the most commonly used absolutely continuous distribution

Exponential growth (e.g. prices, incomes, populations)

• Log-normal distribution, for a single such quantity whose log is normally distributed
• Pareto distribution, for a single such quantity whose log is exponentially distributed; the prototypical power law distribution

Uniformly distributed quantities

• Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair dice)
• Continuous uniform distribution, for absolutely continuously distributed values

Bernoulli trials (yes/no events, with a given probability)

• Basic distributions:
  • Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
  • Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences
  • Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
  • Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution
• Related to sampling schemes over a finite population:
  • Hypergeometric distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using sampling without replacement
  • Beta-binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a Pólya urn model (in some sense, the "opposite" of sampling without replacement)

Categorical outcomes (events with K possible outcomes)

• Categorical distribution, for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the Bernoulli distribution
• Multinomial distribution, for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the binomial distribution
• Multivariate hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the hypergeometric distribution

Poisson process (events that occur independently with a given rate)

• Poisson distribution, for the number of occurrences of a Poisson-type event in a given period of time
• Exponential distribution, for the time before the next Poisson-type event occurs
• Gamma distribution, for the time before the next k Poisson-type events occur

Absolute values of vectors with normally distributed components

• Rayleigh distribution, for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
• Rice distribution, a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in Rician fading of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.

Normally distributed quantities operated with sum of squares

• Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (see chi-squared test)
• Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student's t-test)
• F-distribution, the distribution of the ratio of two scaled chi squared variables; useful e.g. for inferences that involve comparing variances or involving R-squared (the squared correlation coefficient)

As conjugate prior distributions in Bayesian inference

Main article: Conjugate prior

• Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution
• Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a Poisson distribution or exponential distribution, the precision (inverse variance) of a normal distribution, etc.
• Dirichlet distribution, for a vector of probabilities that must sum to 1; conjugate to the categorical distribution and multinomial distribution; generalization of the beta distribution
• Wishart distribution, for a symmetric non-negative definite matrix; conjugate to the inverse of the covariance matrix of a multivariate normal distribution; generalization of the gamma distribution

Some specialized applications of probability distributions

• The cache language models and other statistical language models used in natural language processing to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
• In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point (see Born rule). Therefore, the probability distribution function of the position of a particle is described by $P_{a\le x\le b}(t)={\int }_a^{b}dx|\mathrm{\Psi }(x,t){|}^2$, probability that the particle's position x will be in the interval a ≤ x ≤ b in dimension one, and a similar triple integral in dimension three. This is a key principle of quantum mechanics.
• Probabilistic load flow in power-flow study explains the uncertainties of input variables as probability distribution and provides the power flow calculation also in term of probability distribution.
• Prediction of natural phenomena occurrences based on previous frequency distributions such as tropical cyclones, hail, time in between events, etc.

Fitting

Probability distribution fitting or simply distribution fitting is the fitting of a probability distribution to a series of data concerning the repeated measurement of a variable phenomenon. The aim of distribution fitting is to predict the probability or to forecast the frequency of occurrence of the magnitude of the phenomenon in a certain interval.

There are many probability distributions (see list of probability distributions) of which some can be fitted more closely to the observed frequency of the data than others, depending on the characteristics of the phenomenon and of the distribution. The distribution giving a close fit is supposed to lead to good predictions.

In distribution fitting, therefore, one needs to select a distribution that suits the data well.

Case study

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Case_study

A case study is an in-depth, detailed examination of a particular case (or cases) within a real-world context. For example, case studies in medicine may focus on an individual patient or ailment; case studies in business might cover a particular firm's strategy or a broader market; similarly, case studies in politics can range from a narrow happening over time like the operations of a specific political campaign, to an enormous undertaking like, world war, or more often the policy analysis of real-world problems affecting multiple stakeholders.

Generally, a case study can highlight nearly any individual, group, organization, event, belief system, or action. A case study does not necessarily have to be one observation (N=1), but may include many observations (one or multiple individuals and entities across multiple time periods, all within the same case study). Research projects involving numerous cases are frequently called cross-case research, whereas a study of a single case is called within-case research.

Case study research has been extensively practiced in both the social and natural sciences.

Definition

There are multiple definitions of case studies, which may emphasize the number of observations (a small N), the method (qualitative), the thickness of the research (a comprehensive examination of a phenomenon and its context), and the naturalism (a "real-life context" is being examined) involved in the research. There is general agreement among scholars that a case study does not necessarily have to entail one observation (N=1), but can include many observations within a single case or across numerous cases. For example, a case study of the French Revolution would at the bare minimum be an observation of two observations: France before and after a revolution. John Gerring writes that the N=1 research design is so rare in practice that it amounts to a "myth".

The term cross-case research is frequently used for studies of multiple cases, whereas within-case research is frequently used for a single case study.

John Gerring defines the case study approach as an "intensive study of a single unit or a small number of units (the cases), for the purpose of understanding a larger class of similar units (a population of cases)". According to Gerring, case studies lend themselves to an idiographic style of analysis, whereas quantitative work lends itself to a nomothetic style of analysis. He adds that "the defining feature of qualitative work is its use of noncomparable observations—observations that pertain to different aspects of a causal or descriptive question", whereas quantitative observations are comparable.

According to John Gerring, the key characteristic that distinguishes case studies from all other methods is the "reliance on evidence drawn from a single case and its attempts, at the same time, to illuminate features of a broader set of cases". Scholars use case studies to shed light on a "class" of phenomena.

Research designs

As with other social science methods, no single research design dominates case study research. Case studies can use at least four types of designs. First, there may be a "no theory first" type of case study design, which is closely connected to Kathleen M. Eisenhardt's methodological work. A second type of research design highlights the distinction between single- and multiple-case studies, following Robert K. Yin's guidelines and extensive examples. A third design deals with a "social construction of reality", represented by the work of Robert E. Stake. Finally, the design rationale for a case study may be to identify "anomalies". A representative scholar of this design is Michael Burawoy. Each of these four designs may lead to different applications, and understanding their sometimes unique ontological and epistemological assumptions becomes important. However, although the designs can have substantial methodological differences, the designs also can be used in explicitly acknowledged combinations with each other.

While case studies can be intended to provide bounded explanations of single cases or phenomena, they are often intended to raise theoretical insights about the features of a broader population.

Case selection and structure

Case selection in case study research is generally intended to find cases that are representative samples and which have variations on the dimensions of theoretical interest. Using that is solely representative, such as an average or typical case is often not the richest in information. In clarifying lines of history and causation it is more useful to select subjects that offer an interesting, unusual, or particularly revealing set of circumstances. A case selection that is based on representativeness will seldom be able to produce these kinds of insights.

While a random selection of cases is a valid case selection strategy in large-N research, there is a consensus among scholars that it risks generating serious biases in small-N research. Random selection of cases may produce unrepresentative cases, as well as uninformative cases. Cases should generally be chosen that have a high expected information gain. For example, outlier cases (those which are extreme, deviant or atypical) can reveal more information than the potentially representative case. A case may also be chosen because of the inherent interest of the case or the circumstances surrounding it. Alternatively, it may be chosen because of researchers' in-depth local knowledge; where researchers have this local knowledge they are in a position to "soak and poke" as Richard Fenno put it, and thereby to offer reasoned lines of explanation based on this rich knowledge of setting and circumstances.

Beyond decisions about case selection and the subject and object of the study, decisions need to be made about the purpose, approach, and process of the case study. Gary Thomas thus proposes a typology for the case study wherein purposes are first identified (evaluative or exploratory), then approaches are delineated (theory-testing, theory-building, or illustrative), then processes are decided upon, with a principal choice being between whether the study is to be single or multiple, and choices also about whether the study is to be retrospective, snapshot or diachronic, and whether it is nested, parallel or sequential.

In a 2015 article, John Gerring and Jason Seawright list seven case selection strategies:

1. Typical cases are cases that exemplify a stable cross-case relationship. These cases are representative of the larger population of cases, and the purpose of the study is to look within the case rather than compare it with other cases.
2. Diverse cases are cases that have variations on the relevant X and Y variables. Due to the range of variation on the relevant variables, these cases are representative of the full population of cases.
3. Extreme cases are cases that have an extreme value on the X or Y variable relative to other cases.
4. Deviant cases are cases that defy existing theories and common sense. They not only have extreme values on X or Y (like extreme cases) but defy existing knowledge about causal relations.
5. Influential cases are cases that are central to a model or theory (for example, Nazi Germany in theories of fascism and the far-right).
6. Most similar cases are cases that are similar on all the independent variables, except the one of interest to the researcher.
7. Most different cases are cases that are different on all the independent variables, except the one of interest to the researcher.

For theoretical discovery, Jason Seawright recommends using deviant cases or extreme cases that have an extreme value on the X variable.

Arend Lijphart, and Harry Eckstein identified five types of case study research designs (depending on the research objectives), Alexander George and Andrew Bennett added a sixth category:

1. Atheoretical (or configurative idiographic) case studies aim to describe a case very well, but not to contribute to a theory.
2. Interpretative (or disciplined configurative) case studies aim to use established theories to explain a specific case.
3. Hypothesis-generating (or heuristic) case studies aim to inductively identify new variables, hypotheses, causal mechanisms, and causal paths.
4. Theory testing case studies aim to assess the validity and scope conditions of existing theories.
5. Plausibility probes, aim to assess the plausibility of new hypotheses and theories.
6. Building block studies of types or subtypes, aim to identify common patterns across cases.

Aaron Rapport reformulated "least-likely" and "most-likely" case selection strategies into the "countervailing conditions" case selection strategy. The countervailing conditions case selection strategy has three components:

1. The chosen cases fall within the scope conditions of both the primary theory being tested and the competing alternative hypotheses.
2. For the theories being tested, the analyst must derive clearly stated expected outcomes.
3. In determining how difficult a test is, the analyst should identify the strength of countervailing conditions in the chosen cases.

In terms of case selection, Gary King, Robert Keohane, and Sidney Verba warn against "selecting on the dependent variable". They argue for example that researchers cannot make valid causal inferences about war outbreaks by only looking at instances where war did happen (the researcher should also look at cases where war did not happen). Scholars of qualitative methods have disputed this claim, however. They argue that selecting the dependent variable can be useful depending on the purposes of the research. Barbara Geddes shares their concerns with selecting the dependent variable (she argues that it cannot be used for theory testing purposes), but she argues that selecting on the dependent variable can be useful for theory creation and theory modification.

King, Keohane, and Verba argue that there is no methodological problem in selecting the explanatory variable, however. They do warn about multicollinearity (choosing two or more explanatory variables that perfectly correlate with each other).

Uses

Case studies have commonly been seen as a fruitful way to come up with hypotheses and generate theories. Case studies are useful for understanding outliers or deviant cases. Classic examples of case studies that generated theories includes Darwin's theory of evolution (derived from his travels to the Easter Island), and Douglass North's theories of economic development (derived from case studies of early developing states, such as England).

Case studies are also useful for formulating concepts, which are an important aspect of theory construction. The concepts used in qualitative research will tend to have higher conceptual validity than concepts used in quantitative research (due to conceptual stretching: the unintentional comparison of dissimilar cases). Case studies add descriptive richness, and can have greater internal validity than quantitative studies. Case studies are suited to explain outcomes in individual cases, which is something that quantitative methods are less equipped to do.

Case studies have been characterized as useful to assess the plausibility of arguments that explain empirical regularities. Case studies are also useful for understanding outliers or deviant cases.

Through fine-gained knowledge and description, case studies can fully specify the causal mechanisms in a way that may be harder in a large-N study. In terms of identifying "causal mechanisms", some scholars distinguish between "weak" and "strong chains". Strong chains actively connect elements of the causal chain to produce an outcome whereas weak chains are just intervening variables.

Case studies of cases that defy existing theoretical expectations may contribute knowledge by delineating why the cases violate theoretical predictions and specifying the scope conditions of the theory. Case studies are useful in situations of causal complexity where there may be equifinality, complex interaction effects and path dependency. They may also be more appropriate for empirical verifications of strategic interactions in rationalist scholarship than quantitative methods. Case studies can identify necessary and insufficient conditions, as well as complex combinations of necessary and sufficient conditions. They argue that case studies may also be useful in identifying the scope conditions of a theory: whether variables are sufficient or necessary to bring about an outcome.

Qualitative research may be necessary to determine whether a treatment is as-if random or not. As a consequence, good quantitative observational research often entails a qualitative component.

Limitations

Designing Social Inquiry (also called "KKV"), an influential 1994 book written by Gary King, Robert Keohane, and Sidney Verba, primarily applies lessons from regression-oriented analysis to qualitative research, arguing that the same logics of causal inference can be used in both types of research. The authors' recommendation is to increase the number of observations (a recommendation that Barbara Geddes also makes in Paradigms and Sand Castles), because few observations make it harder to estimate multiple causal effects, as well as increase the risk that there is measurement error, and that an event in a single case was caused by random error or unobservable factors. KKV sees process-tracing and qualitative research as being "unable to yield strong causal inference" due to the fact that qualitative scholars would struggle with determining which of many intervening variables truly links the independent variable with a dependent variable. The primary problem is that qualitative research lacks a sufficient number of observations to properly estimate the effects of an independent variable. They write that the number of observations could be increased through various means, but that would simultaneously lead to another problem: that the number of variables would increase and thus reduce degrees of freedom. Christopher H. Achen and Duncan Snidal similarly argue that case studies are not useful for theory construction and theory testing.

The purported "degrees of freedom" problem that KKV identify is widely considered flawed; while quantitative scholars try to aggregate variables to reduce the number of variables and thus increase the degrees of freedom, qualitative scholars intentionally want their variables to have many different attributes and complexity. For example, James Mahoney writes, "the Bayesian nature of process of tracing explains why it is inappropriate to view qualitative research as suffering from a small-N problem and certain standard causal identification problems." By using Bayesian probability, it may be possible to makes strong causal inferences from a small sliver of data.

KKV also identify inductive reasoning in qualitative research as a problem, arguing that scholars should not revise hypotheses during or after data has been collected because it allows for ad hoc theoretical adjustments to fit the collected data. However, scholars have pushed back on this claim, noting that inductive reasoning is a legitimate practice (both in qualitative and quantitative research).

A commonly described limit of case studies is that they do not lend themselves to generalizability. Due to the small number of cases, it may be harder to ensure that the chosen cases are representative of the larger population. Some scholars, such as Bent Flyvbjerg, have pushed back on that notion.

As small-N research should not rely on random sampling, scholars must be careful in avoiding selection bias when picking suitable cases. A common criticism of qualitative scholarship is that cases are chosen because they are consistent with the scholar's preconceived notions, resulting in biased research. Alexander George and Andrew Bennett also note that a common problem in case study research is that of reconciling conflicting interpretations of the same data. Another limit of case study research is that it can be hard to estimate the magnitude of causal effects.

Teaching case studies

Teachers may prepare a case study that will then be used in classrooms in the form of a "teaching" case study (also see case method and casebook method). For instance, as early as 1870 at Harvard Law School, Christopher Langdell departed from the traditional lecture-and-notes approach to teaching contract law and began using cases pled before courts as the basis for class discussions. By 1920, this practice had become the dominant pedagogical approach used by law schools in the United States.

Engineering students participate in a case study competition.

Outside of law, teaching case studies have become popular in many different fields and professions, ranging from business education to science education. The Harvard Business School has been among the most prominent developers and users of teaching case studies. Teachers develop case studies with particular learning objectives in mind. Additional relevant documentation, such as financial statements, time-lines, short biographies, and multimedia supplements (such as video-recordings of interviews) often accompany the case studies. Similarly, teaching case studies have become increasingly popular in science education, covering different biological and physical sciences. The National Center for Case Studies in Teaching Science has made a growing body of teaching case studies available for classroom use, for university as well as secondary school coursework.