Many probability distributions that are important in theory or applications have been given specific names.
YouTube Encyclopedic

1/5Views:123 5751 671196 500750 0654 465

✪ Overview of Some Discrete Probability Distributions (Binomial,Geometric,Hypergeometric,Poisson,NegB)

✪ Probability: Types of Distributions

✪ The Shape of Data: Distributions: Crash Course Statistics #7

✪ Constructing a probability distribution for random variable  Khan Academy

✪ 2 Types of Probability Distribution
Transcription
Let's look at a quick overview of some discrete probability distributions and their relationships. I intend this video to be used as a recap after having been introduced to these distributions, but it could possibly be used as an introductory overview. I don't any calculations in this video, nor do I discuss how to calculate the probabilities. I simply discuss how these different distributions arise, and the relationships between them. The Bernoulli distribution is the distribution of the number of successes on a single Bernoulli trial. In a Bernoulli trial we get either a success or a failure. It's like an answer to a yes or no question. A Bernoulli random variable can take on only the values 0 and 1. For example, we can use the Bernoulli distribution to answer questions like: if a single coin is tossed once, what is the probability it comes up heads? Or, if a single adult American is randomly selected, what is the probability they are a heart surgeon? Some other important distributions are built on the notion of independent Bernoulli trials, where we have a series of trials, and each one results in a success or a failure. An important one is the binomial distribution, which is the distribution of the number of successes in n independent Bernoulli trials. For example, with the binomial distribution we can answer a question like: if a coin is tossed 20 times, what is the probability heads comes up exactly 14 times? And since the binomial distribution is the distribution of the number of successes in n independent Bernoulli trials, the Bernoulli distribution is a special case of the binomial distribution with n=1, a single trial. Continuing on with the theme of independent Bernoulli trials, the geometric distribution is the distribution of the number of trials needed to get the first success. For example, with the geometric distribution we can answer a question like: if a coin has repeatedly tossed, what is the probability the first time heads appears occurs on the 8 toss? The negative binomial distribution is a generalization of the geometric distribution. The negative binomial distribution is the distribution of the number of trials needed to get a certain number of successes in repeated independent Bernoulli trials. So the negative binomial distribution can help us answer questions like: if a coin has repeatedly tossed, what is the probability the third time heads appears occurs on the ninth trial? The way the binomial distribution and the negative binomial distribution arise can sound similar, and they can sometimes be confused. They differ in what the random variable is. In the binomial distribution, the number of trials is fixed, and the number of successes is a random variable. For instance, we're tossing a coin a fixed number of times, and the number of heads that comes up is a random variable. In the negative binomial distribution, the number of successes is fixed, and the number of trials required to get that number of successes is the random variable. For instance, we might be tossing a coin until we get heads 4 times. And the number of tosses required to get heads 4 times is the random variable. Now I'll talk about two distributions that are related to the binomial, but aren't based on independent Bernoulli trials. The hypergeometric distribution is similar to the binomial distribution in that we're interested in the number of successes in n trials, but it's different because the trials are not independent. The hypergeometric distribution is the distribution of the number of successes when we are drawing without replacement from a source that contains a certain number of successes and a certain number of failures. For example, we can use the hypergeometric distribution to answer a question like: if 5 cards are drawn without replacement from a well shuffled deck, what is the probability exactly 3 hearts are drawn? It's different from the binomial because the probability of success, the probability of getting a heart, would change from card to card, depending on what happened before. However, if the cards are drawn with replacement, meaning the card was put back in and reshuffled before the next card was drawn, then the trials would be independent and we would use the binomial distribution instead. If we are sampling only a small fraction of objects without replacement from a large population then the trials are still not independent, but that dependency has only a small effect, and the binomial distribution closely approximates the hypergeometric distribution. So there are times when a problem is in its nature a hypergeometric problem, but we use the binomial distribution as an approximation. This can make our life a little bit easier sometimes. Another distribution related to the binomial is the Poisson distribution. But this one's a little harder to explain. The Poisson distribution is the distribution of the number of events in a given time or length, or area, or volume etc., if those events are occurring randomly and independently. There's a bit more to it than that, and I go into this in much greater detail in my Poisson videos. But that's the gist of it. So we might use the Poisson distribution to answer a question like: what is the probability there will be exactly 4 car accidents on a certain university campus in a given week? There is a relationship between the Poisson distribution and the binomial distribution. The Poisson distribution closely approximates the binomial distribution if n, the number of trials, in the binomial, is large and p, the probability of success, is very small. So suppose we have a question like: what is the probability that in a random sample 100,000 births, there is at least one case of progeria? Progeria is an extremely rare disease that causes premature aging, and it occurs in about 1 in every eight million births. This is truly a binomial problem. But we have a binomial problem with a very large n, 100,000, and a very small probability of success, 1 in eight million or so, because progeria such a rare disease. And so this could be very well approximated by the Poisson distribution. I look into all of these concepts discussed in this video in greater detail in the videos for these specific distributions.
Contents
Discrete distributions
With finite support
 The Bernoulli distribution, which takes value 1 with probability p and value 0 with probability q = 1 − p.
 The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
 The binomial distribution, which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success.
 The betabinomial distribution, which describes the number of successes in a series of independent Yes/No experiments with heterogeneity in the success probability.
 The degenerate distribution at x_{0}, where X is certain to take the value x_{0}. This does not look random, but it satisfies the definition of random variable. This is useful because it puts deterministic variables and random variables in the same formalism.
 The discrete uniform distribution, where all elements of a finite set are equally likely. This is the theoretical distribution model for a balanced coin, an unbiased die, a casino roulette, or the first card of a wellshuffled deck.
 The hypergeometric distribution, which describes the number of successes in the first m of a series of n consecutive Yes/No experiments, if the total number of successes is known. This distribution arises when there is no replacement.
 The Poisson binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with different success probabilities.
 Fisher's noncentral hypergeometric distribution
 Wallenius' noncentral hypergeometric distribution
 Benford's law, which describes the frequency of the first digit of many naturally occurring data.
 The ideal and robust soliton distributions.
With infinite support
 The beta negative binomial distribution
 The Boltzmann distribution, a discrete distribution important in statistical physics which describes the probabilities of the various discrete energy levels of a system in thermal equilibrium. It has a continuous analogue. Special cases include:
 The Borel distribution
 The extended negative binomial distribution
 The extended hypergeometric distribution
 The generalized logseries distribution
 The geometric distribution, a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Bernoulli trials, or alternatively only the number of losses before the first success (i.e. one less).
 The logarithmic (series) distribution
 The negative binomial distribution or Pascal distribution, a generalization of the geometric distribution to the nth success.
 The discrete compound Poisson distribution
 The parabolic fractal distribution
 The Poisson distribution, which describes a very large number of individually unlikely events that happen in a certain time interval. Related to this distribution are a number of other distributions: the displaced Poisson, the hyperPoisson, the general Poisson binomial and the Poisson type distributions.
 The Conway–Maxwell–Poisson distribution, a twoparameter extension of the Poisson distribution with an adjustable rate of decay.
 The Zerotruncated Poisson distribution, for processes in which zero counts are not observed
 The Polya–Eggenberger distribution
 The Skellam distribution, the distribution of the difference between two independent Poissondistributed random variables.
 The skew elliptical distribution
 The Yule–Simon distribution
 The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is the Zipf distribution for an infinite number of elements.
 Zipf's law or the Zipf distribution. A discrete powerlaw distribution, the most famous example of which is the description of the frequency of words in the English language.
 The Zipf–Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution.
Continuous distributions
Supported on a bounded interval
 The arcsine distribution on [a,b], which is a special case of the Beta distribution if α=β=1/2, a=0, and b = 1.
 The Beta distribution on [0,1], a family of twoparameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities.
 The logitnormal distribution on (0,1).
 The Dirac delta function although not strictly a distribution, is a limiting form of many continuous probability functions. It represents a discrete probability distribution concentrated at 0 — a degenerate distribution — but the notation treats it as if it were a continuous distribution.
 The uniform distribution or rectangular distribution on [a,b], where all points in a finite interval are equally likely.
 The Irwin–Hall distribution is the distribution of the sum of n independent random variables, each of which having the uniform distribution on [0,1].
 The Bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1].
 The Kent distribution on the threedimensional sphere.
 The Kumaraswamy distribution is as versatile as the Beta distribution but has simple closed forms for both the cdf and the pdf.
 The Marchenko–Pastur distribution is important in the theory of random matrices.
 The PERT distribution is a special case of the beta distribution
 The raised cosine distribution on []
 The reciprocal distribution
 The triangular distribution on [a, b], a special case of which is the distribution of the sum of two independent uniformly distributed random variables (the convolution of two uniform distributions).
 The trapezoidal distribution
 The truncated normal distribution on [a, b].
 The Uquadratic distribution on [a, b].
 The von Mises–Fisher distribution on the Ndimensional sphere has the von Mises distribution as a special case.
 The Wigner semicircle distribution is important in the theory of random matrices.
Supported on intervals of length 2π – directional distributions
 The HenyeyGreenstein phase function
 The Mie phase function
 The von Mises distribution
 The wrapped normal distribution
 The wrapped exponential distribution
 The wrapped Lévy distribution
 The wrapped Cauchy distribution
 The wrapped Laplace distribution
 The wrapped asymmetric Laplace distribution
 The Dirac comb of period 2 π although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrapped Dirac delta function. It represents a discrete probability distribution concentrated at 2πn — a degenerate distribution — but the notation treats it as if it were a continuous distribution.
Supported on semiinfinite intervals, usually [0,∞)
 The Beta prime distribution
 The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times.
 The chi distribution
 The chisquared distribution, which is the sum of the squares of n independent Gaussian random variables. It is a special case of the Gamma distribution, and it is used in goodnessoffit tests in statistics.
 The Dagum distribution
 The exponential distribution, which describes the time between consecutive rare random events in a process with no memory.
 The Exponentiallogarithmic distribution
 The Fdistribution, which is the distribution of the ratio of two (normalized) chisquareddistributed random variables, used in the analysis of variance. It is referred to as the beta prime distribution when it is the ratio of two chisquared variates which are not normalized by dividing them by their numbers of degrees of freedom.
 The folded normal distribution
 The Fréchet distribution
 The Gamma distribution, which describes the time until n consecutive rare random events occur in a process with no memory.
 The Erlang distribution, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times in queuing systems
 The inversegamma distribution
 The Generalized gamma distribution
 The generalized Pareto distribution
 The Gamma/Gompertz distribution
 The Gompertz distribution
 The halfnormal distribution
 Hotelling's Tsquared distribution
 The inverse Gaussian distribution, also known as the Wald distribution
 The Lévy distribution
 The logCauchy distribution
 The logLaplace distribution
 The loglogistic distribution
 The lognormal distribution, describing variables which can be modelled as the product of many small independent positive variables.
 The Lomax distribution
 The MittagLeffler distribution
 The Nakagami distribution
 The Pareto distribution, or "power law" distribution, used in the analysis of financial data and critical behavior.
 The Pearson Type III distribution
 The Phasetype distribution, used in queueing theory
 The phased biexponential distribution is commonly used in pharmacokinetics
 The phased biWeibull distribution
 The Rayleigh distribution
 The Rayleigh mixture distribution
 The Rice distribution
 The shifted Gompertz distribution
 The type2 Gumbel distribution
 The Weibull distribution or Rosin Rammler distribution, of which the exponential distribution is a special case, is used to model the lifetime of technical devices and is used to describe the particle size distribution of particles generated by grinding, milling and crushing operations.
Supported on the whole real line
 The Behrens–Fisher distribution, which arises in the Behrens–Fisher problem.
 The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian profile, and is associated with many processes, including resonance energy distribution, impact and natural spectral line broadening and quadratic stark line broadening.
 Chernoff's distribution
 The Exponentially modified Gaussian distribution, a convolution of a normal distribution with an exponential distribution, and the Gaussian minus exponential distribution, a convolution of a normal distribution with the negative of an exponential distribution.
 The Fisher–Tippett, extreme value, or logWeibull distribution
 Fisher's zdistribution
 The skewed generalized t distribution
 The generalized logistic distribution
 The generalized normal distribution
 The geometric stable distribution
 The Gumbel distribution
 The Holtsmark distribution, an example of a distribution that has a finite expected value but infinite variance.
 The hyperbolic distribution
 The hyperbolic secant distribution
 The Johnson SU distribution
 The Landau distribution
 The Laplace distribution
 The Lévy skew alphastable distribution or stable distribution is a family of distributions often used to characterize financial data and critical behavior; the Cauchy distribution, Holtsmark distribution, Landau distribution, Lévy distribution and normal distribution are special cases.
 The Linnik distribution
 The logistic distribution
 The mapAiry distribution
 The normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent, identically distributed variables with finite mean and variance is approximately normal.
 The Normalexponentialgamma distribution
 The Normalinverse Gaussian distribution
 The Pearson Type IV distribution (see Pearson distributions)
 The skew normal distribution
 Student's tdistribution, useful for estimating unknown means of Gaussian populations.
 The noncentral tdistribution
 The skew t distribution
 The Champernowne distribution
 The type1 Gumbel distribution
 The Tracy–Widom distribution
 The Voigt distribution, or Voigt profile, is the convolution of a normal distribution and a Cauchy distribution. It is found in spectroscopy when spectral line profiles are broadened by a mixture of Lorentzian and Doppler broadening mechanisms.
 The Chen distribution.
With variable support
 The generalized extreme value distribution has a finite upper bound or a finite lower bound depending on what range the value of one of the parameters of the distribution is in (or is supported on the whole real line for one special value of the parameter
 The generalized Pareto distribution has a support which is either bounded below only, or bounded both above and below
 The Tukey lambda distribution is either supported on the whole real line, or on a bounded interval, depending on what range the value of one of the parameters of the distribution is in.
 The Wakeby distribution
Mixed discrete/continuous distributions
 The rectified Gaussian distribution replaces negative values from a normal distribution with a discrete component at zero.
 The compound poissongamma or Tweedie distribution is continuous over the strictly positive real numbers, with a mass at zero.
Joint distributions
For any set of independent random variables the probability density function of their joint distribution is the product of their individual density functions.
Two or more random variables on the same sample space
 The Dirichlet distribution, a generalization of the beta distribution.
 The Ewens's sampling formula is a probability distribution on the set of all partitions of an integer n, arising in population genetics.
 The Balding–Nichols model
 The multinomial distribution, a generalization of the binomial distribution.
 The multivariate normal distribution, a generalization of the normal distribution.
 The multivariate tdistribution, a generalization of the Student's tdistribution.
 The negative multinomial distribution, a generalization of the negative binomial distribution.
 The generalized multivariate loggamma distribution
 The Marshall–Olkin exponential distribution
Distributions of matrixvalued random variables
 The Wishart distribution
 The inverseWishart distribution
 The matrix normal distribution
 The matrix tdistribution
Nonnumeric distributions
Miscellaneous distributions
 The Cantor distribution
 The generalized logistic distribution family
 The Pearson distribution family
 The phasetype distribution
See also
 Mixture distribution
 Cumulative distribution function
 Likelihood function
 List of statistical topics
 Probability density function
 Random variable
 Histogram
 Truncated distribution
 Copula (statistics)
 Probability distribution
 Relationships among probability distributions
 ProbOnto a knowledge base and ontology of probability distributions, URL: probonto.org