In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. The distribution simplifies when c = a or c = b and this can be obtained from the cumulative distribution function. It is based on a knowledge of the minimum and maximum, for these reasons, the triangle distribution has been called a lack of knowledge distribution. The triangular distribution is often used in business decision making. Generally, when not much is known about the distribution of an outcome, but if the most likely outcome is known, the outcome can be simulated by a triangular distribution. See for example under corporate finance, the triangular distribution, along with the Beta distribution, is widely used in project management to model events which take place within an interval defined by a minimum and maximum value. The symmetric triangular distribution is used in audio dithering, where it is called TPDF.
Trapezoidal distribution Thomas Simpson Three-point estimation Five-number summary Seven-number summary Triangular function Weisstein, triangle Distribution, decisionsciences. org Triangular Distribution, brighton-webs. co. uk
For instance, if the random variable X is used to denote the outcome of a coin toss, the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. In more technical terms, the probability distribution is a description of a phenomenon in terms of the probabilities of events. Examples of random phenomena can include the results of an experiment or survey, a probability distribution is defined in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of numbers or a higher-dimensional vector space, or it may be a list of non-numerical values, for example. Probability distributions are divided into two classes. A discrete probability distribution can be encoded by a discrete list of the probabilities of the outcomes, on the other hand, a continuous probability distribution is typically described by probability density functions. The normal distribution represents a commonly encountered 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 the set of numbers is called univariate.
Important and commonly encountered univariate probability distributions include the distribution, the hypergeometric distribution. The multivariate normal distribution is a commonly encountered multivariate distribution, to define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables. For example, the probability that an object weighs exactly 500 g is zero. Continuous probability distributions can be described in several ways, the cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. As probability theory is used in diverse applications, terminology is not uniform. The following terms are used for probability distribution functions, Distribution. Probability distribution, is a table that displays the probabilities of outcomes in a sample. Could be called a frequency distribution table, where all occurrences of outcomes sum to 1. Distribution function, is a form of frequency distribution table.
Probability distribution function, is a form of probability distribution table
In probability theory and statistics, the chi-squared distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. When it is being distinguished from the more general noncentral chi-squared distribution, many other statistical tests use this distribution, like Friedmans analysis of variance by ranks. Zk are independent, standard normal variables, the sum of their squares. This is usually denoted as Q ∼ χ2 or Q ∼ χ k 2, the chi-squared distribution has one parameter, k — a positive integer that specifies the number of degrees of freedom The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known such as the normal distribution and the exponential distribution. It arises in the following tests, among others. The primary reason that the distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t statistic in a t-test, for these hypothesis tests, as the sample size, n, the sampling distribution of the test statistic approaches the normal distribution.
Testing hypotheses using a distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a normal distribution. So wherever a normal distribution could be used for a hypothesis test, suppose that Z is a standard normal random variable, with mean =0 and variance =1. A sample drawn at random from Z is a sample from the shown in the graph of the standard normal distribution. Define a new random variable Q, to generate a random sample from Q, take a sample from Z and square the value. The distribution of the values is given by the random variable Q = Z2. The distribution of the random variable Q is an example of a chi-squared distribution, the subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single normal distribution is said to have 1 degree of freedom. Just as extreme values of the distribution have low probability. An additional reason that the distribution is widely used is that it is a member of the class of likelihood ratio tests
The Poisson distribution can be used for the number of events in other specified intervals such as distance, area or volume. For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive a number of 4 letters per day. Other examples that may follow a Poisson, the number of calls received by a call center per hour or the number of decay events per second from a radioactive source. The Poisson distribution is popular for modelling the number of times an event occurs in an interval of time or space. K is the number of times an event occurs in an interval, the rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals, two events cannot occur at exactly the same instant. The probability of an event in an interval is proportional to the length of the interval. If these conditions are true, K is a Poisson random variable, an event can occur 0,1,2, … times in an interval. The average number of events in an interval is designated λ, lambda is the event rate, called the rate parameter.
The probability of observing k events in an interval is given by the equation P = e − λ λ k k. where λ is the number of events per interval e is the number 2.71828. The base of the natural logarithms k takes values 0,1,2, … k, = k × × × … ×2 ×1 is the factorial of k. This equation is the probability function for a Poisson distribution. On a particular river, overflow floods occur once every 100 years on average, calculate the probability of k =0,1,2,3,4,5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate. Because the average event rate is one overflow flood per 100 years, = e −11 =0.368 P =11 e −11. = e −11 =0.368 P =12 e −12, = e −12 =0.184 The table below gives the probability for 0 to 6 overflow floods in a 100-year period. Ugarte and colleagues report that the number of goals in a World Cup soccer match is approximately 2.5. Because the average event rate is 2.5 goals per match, = e −2.51 =0.082 P =2.51 e −2.51. =2.5 e −2.51 =0.205 P =2.52 e −2.52
Discrete uniform distribution
Another way of saying discrete uniform distribution would be a known, finite number of outcomes equally likely to happen. A simple example of the uniform distribution is throwing a fair die. The possible values are 1,2,3,4,5,6, if two dice are thrown and their values added, the resulting distribution is no longer uniform since not all sums have equal probability. The discrete uniform distribution itself is inherently non-parametric and it is convenient, however, to represent its values generally by an integer interval, so that a, b become the main parameters of the distribution. This problem is known as the German tank problem, following the application of maximum estimation to estimates of German tank production during World War II. The UMVU estimator for the maximum is given by N ^ = k +1 k m −1 = m + m k −1 where m is the maximum and k is the sample size. This can be seen as a simple case of maximum spacing estimation. This has a variance of 1 k ≈ N2 k 2 for small samples k ≪ N so a standard deviation of approximately N k, the sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased.
If samples are not numbered but are recognizable or markable, one can instead estimate population size via the capture-recapture method, see rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation
These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the one, however, to avoid ambiguity, it is considered wise to indicate which is intended. It’s the probability that the first occurrence of success requires k number of independent trials, each with success probability p. If the probability of success on each trial is p, the probability that the kth trial is the first success is Pr = k −1 p for k =1,2,3. The above form of distribution is used for modeling the number of trials up to. By contrast, the form of the geometric distribution is used for modeling the number of failures until the first success. In either case, the sequence of probabilities is a geometric sequence, for example, suppose an ordinary die is thrown repeatedly until the first time a 1 appears. The probability distribution of the number of times it is thrown is supported on the set and is a geometric distribution with p = 1/6. Consider a sequence of trials, where each trial has only two possible outcomes, the probability of success is assumed to be the same for each trial.
In such a sequence of trials, the distribution is useful to model the number of failures before the first success. The distribution gives the probability that there are zero failures before the first success, one failure before the first success, a newly-wed couple plans to have children, and will continue until the first girl. What is the probability that there are zero boys before the first girl, one boy before the first girl, a doctor is seeking an anti-depressant for a newly diagnosed patient. Suppose that, of the available anti-depressant drugs, the probability that any particular drug will be effective for a patient is p=0.6. What is the probability that the first drug found to be effective for this patient is the first drug tried, the second drug tried, what is the expected number of drugs that will be tried to find one that is effective. A patient is waiting for a suitable matching kidney donor for a transplant, if the probability that a randomly selected donor is a suitable match is p=0.1, what is the expected number of donors who will be tested before a matching donor is found.
The geometric distribution is a model if the following assumptions are true. The phenomenon being modelled is a sequence of independent trials, there are only two possible outcomes for each trial, often designated success or failure. The probability of success, p, is the same for every trial, if these conditions are true, the geometric random variable is the count of the number of failures before the first success
The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined using the convolution of a binomial distribution with a Poisson distribution. The skewness of the Delaporte distribution is, λ + α β32 The excess kurtosis of the distribution is, λ +3 λ2 + α β2 Murat, M. Szynal, on moments of counting distributions satisfying the kth-order recursion and their compound distributions
Benfords law, called the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in naturally occurring collections of numbers. For example, in sets which obey the law, the number 1 appears as the most significant digit about 30% of the time, by contrast, if the digits were distributed uniformly, they would each occur about 11. 1% of the time. Benfords law makes predictions about the distribution of digits, third digits, digit combinations. It tends to be most accurate values are distributed across multiple orders of magnitude. The graph here shows Benfords law for base 10, there is a generalization of the law to numbers expressed in other bases, and a generalization from leading 1 digit to leading n digits. It is named after physicist Frank Benford, who stated it in 1938, Benfords law is a special case of Zipfs law. A set of numbers is said to satisfy Benfords law if the digit d occurs with probability P = log 10 − log 10 = log 10 = log 10 .
Therefore, this is the distribution expected if the mantissae of the logarithms of the numbers are uniformly and randomly distributed. For example, a x, constrained to lie between 1 and 10, starts with the digit 1 if 1 ≤ x <2. Therefore, x starts with the digit 1 if log 1 ≤ log x < log 2, the probabilities are proportional to the interval widths, and this gives the equation above. An extension of Benfords law predicts the distribution of first digits in other bases besides decimal, in fact, the general form is, P = log b − log b = log b . For b =2, Benfords law is true but trivial, the discovery of Benfords law goes back to 1881, when the American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages were much more worn than the other pages. Newcombs published result is the first known instance of this observation and includes a distribution on the second digit, Newcomb proposed a law that the probability of a single number N being the first digit of a number was equal to log − log.
The phenomenon was noted in 1938 by the physicist Frank Benford. The total number of used in the paper was 20,229. This discovery was named after Benford. In 1995, Ted Hill proved the result about mixed distributions mentioned below, arno Berger and Ted Hill have stated that, The widely known phenomenon called Benford’s law continues to defy attempts at an easy derivation
In probability and statistics, the logarithmic distribution is a discrete probability distribution derived from the Maclaurin series expansion − ln = p + p 22 + p 33 + ⋯. From this we obtain the identity ∑ k =1 ∞ −1 ln p k k =1. This leads directly to the probability function of a Log-distributed random variable, f = −1 ln p k k for k ≥1. Because of the identity above, the distribution is properly normalized, the cumulative distribution function is F =1 + B ln where B is the incomplete beta function. A Poisson compounded with Log-distributed random variables has a binomial distribution. In this way, the binomial distribution is seen to be a compound Poisson distribution. R. A. Fisher described the distribution in a paper that used it to model relative species abundance. The probability mass function ƒ of this distribution satisfies the relation f = k p k +1 f. Poisson distribution Johnson, Norman Lloyd, Adrienne W, chapter 7, Logarithmic and Lagrangian distributions