Handbook on Probability Distributions
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A Random Variable X with Pdf G(X) = Λα Γ(Α) X ≥ 0 Has Gamma
DISTRIBUTIONS DERIVED FROM THE NORMAL DISTRIBUTION Definition: A random variable X with pdf λα g(x) = xα−1e−λx x ≥ 0 Γ(α) has gamma distribution with parameters α > 0 and λ > 0. The gamma function Γ(x), is defined as Z ∞ Γ(x) = ux−1e−udu. 0 Properties of the Gamma Function: (i) Γ(x + 1) = xΓ(x) (ii) Γ(n + 1) = n! √ (iii) Γ(1/2) = π. Remarks: 1. Notice that an exponential rv with parameter 1/θ = λ is a special case of a gamma rv. with parameters α = 1 and λ. 2. The sum of n independent identically distributed (iid) exponential rv, with parameter λ has a gamma distribution, with parameters n and λ. 3. The sum of n iid gamma rv with parameters α and λ has gamma distribution with parameters nα and λ. Definition: If Z is a standard normal rv, the distribution of U = Z2 called the chi-square distribution with 1 degree of freedom. 2 The density function of U ∼ χ1 is −1/2 x −x/2 fU (x) = √ e , x > 0. 2π 2 Remark: A χ1 random variable has the same density as a random variable with gamma distribution, with parameters α = 1/2 and λ = 1/2. Definition: If U1,U2,...,Uk are independent chi-square rv-s with 1 degree of freedom, the distribution of V = U1 + U2 + ... + Uk is called the chi-square distribution with k degrees of freedom. 2 Using Remark 3. and the above remark, a χk rv. follows gamma distribution with parameters 2 α = k/2 and λ = 1/2. -
Arxiv:1906.06684V1 [Math.NA]
Randomized Computation of Continuous Data: Is Brownian Motion Computable?∗ Willem L. Fouch´e1, Hyunwoo Lee2, Donghyun Lim2, Sewon Park2, Matthias Schr¨oder3, Martin Ziegler2 1 University of South Africa 2 KAIST 3 University of Birmingham Abstract. We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite fair coin flips. This extends [Schr¨oder&Simpson’05] and [Hoyrup&Rojas’09] considering sequences of suitably and adaptively biased coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a prob- ability distribution on the space of continuous functions f : [0, 1] → R with f(0) = 0 whose computability has been conjectured [Davie&Fouch´e’2013; arXiv:1409.4667,§6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure. 1 Introduction Randomization is a powerful technique in classical (i.e. discrete) Computer Science: Many difficult problems have turned out to admit simple solutions by algorithms that ‘roll dice’ and are efficient/correct/optimal with high probability [DKM+94,BMadHS99,CS00,BV04]. Indeed, fair coin flips have been shown computationally universal [Wal77]. Over continuous data, well-known closely connected to topology [Grz57] [Wei00, 2.2+ 3], notions of proba- § § bilistic computation are more subtle [BGH15,Col15]. 1.1 Overview Section 2 resumes from [SS06] the question of how to represent Borel probability measures. -
Connection Between Dirichlet Distributions and a Scale-Invariant Probabilistic Model Based on Leibniz-Like Pyramids
Home Search Collections Journals About Contact us My IOPscience Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids This content has been downloaded from IOPscience. Please scroll down to see the full text. View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 152.84.125.254 This content was downloaded on 22/12/2014 at 23:17 Please note that terms and conditions apply. Journal of Statistical Mechanics: Theory and Experiment Connection between Dirichlet distributions and a scale-invariant probabilistic model based on J. Stat. Mech. Leibniz-like pyramids A Rodr´ıguez1 and C Tsallis2,3 1 Departamento de Matem´atica Aplicada a la Ingenier´ıa Aeroespacial, Universidad Polit´ecnica de Madrid, Pza. Cardenal Cisneros s/n, 28040 Madrid, Spain 2 Centro Brasileiro de Pesquisas F´ısicas and Instituto Nacional de Ciˆencia e Tecnologia de Sistemas Complexos, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil (2014) P12027 3 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA E-mail: [email protected] and [email protected] Received 4 November 2014 Accepted for publication 30 November 2014 Published 22 December 2014 Online at stacks.iop.org/JSTAT/2014/P12027 doi:10.1088/1742-5468/2014/12/P12027 Abstract. We show that the N limiting probability distributions of a recently introduced family of d→∞-dimensional scale-invariant probabilistic models based on Leibniz-like (d + 1)-dimensional hyperpyramids (Rodr´ıguez and Tsallis 2012 J. Math. Phys. 53 023302) are given by Dirichlet distributions for d = 1, 2, ... -
Stat 5101 Notes: Brand Name Distributions
Stat 5101 Notes: Brand Name Distributions Charles J. Geyer February 14, 2003 1 Discrete Uniform Distribution Symbol DiscreteUniform(n). Type Discrete. Rationale Equally likely outcomes. Sample Space The interval 1, 2, ..., n of the integers. Probability Function 1 f(x) = , x = 1, 2, . , n n Moments n + 1 E(X) = 2 n2 − 1 var(X) = 12 2 Uniform Distribution Symbol Uniform(a, b). Type Continuous. Rationale Continuous analog of the discrete uniform distribution. Parameters Real numbers a and b with a < b. Sample Space The interval (a, b) of the real numbers. 1 Probability Density Function 1 f(x) = , a < x < b b − a Moments a + b E(X) = 2 (b − a)2 var(X) = 12 Relation to Other Distributions Beta(1, 1) = Uniform(0, 1). 3 Bernoulli Distribution Symbol Bernoulli(p). Type Discrete. Rationale Any zero-or-one-valued random variable. Parameter Real number 0 ≤ p ≤ 1. Sample Space The two-element set {0, 1}. Probability Function ( p, x = 1 f(x) = 1 − p, x = 0 Moments E(X) = p var(X) = p(1 − p) Addition Rule If X1, ..., Xk are i. i. d. Bernoulli(p) random variables, then X1 + ··· + Xk is a Binomial(k, p) random variable. Relation to Other Distributions Bernoulli(p) = Binomial(1, p). 4 Binomial Distribution Symbol Binomial(n, p). 2 Type Discrete. Rationale Sum of i. i. d. Bernoulli random variables. Parameters Real number 0 ≤ p ≤ 1. Integer n ≥ 1. Sample Space The interval 0, 1, ..., n of the integers. Probability Function n f(x) = px(1 − p)n−x, x = 0, 1, . , n x Moments E(X) = np var(X) = np(1 − p) Addition Rule If X1, ..., Xk are independent random variables, Xi being Binomial(ni, p) distributed, then X1 + ··· + Xk is a Binomial(n1 + ··· + nk, p) random variable. -
Modelling Censored Losses Using Splicing: a Global Fit Strategy With
Modelling censored losses using splicing: a global fit strategy with mixed Erlang and extreme value distributions Tom Reynkens∗a, Roel Verbelenb, Jan Beirlanta,c, and Katrien Antoniob,d aLStat and LRisk, Department of Mathematics, KU Leuven. Celestijnenlaan 200B, 3001 Leuven, Belgium. bLStat and LRisk, Faculty of Economics and Business, KU Leuven. Naamsestraat 69, 3000 Leuven, Belgium. cDepartment of Mathematical Statistics and Actuarial Science, University of the Free State. P.O. Box 339, Bloemfontein 9300, South Africa. dFaculty of Economics and Business, University of Amsterdam. Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. August 14, 2017 Abstract In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the arXiv:1608.01566v4 [stat.ME] 11 Aug 2017 flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data. Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement. -
Distribution of Mutual Information
Distribution of Mutual Information Marcus Hutter IDSIA, Galleria 2, CH-6928 Manno-Lugano, Switzerland [email protected] http://www.idsia.ch/-marcus Abstract The mutual information of two random variables z and J with joint probabilities {7rij} is commonly used in learning Bayesian nets as well as in many other fields. The chances 7rij are usually estimated by the empirical sampling frequency nij In leading to a point es timate J(nij In) for the mutual information. To answer questions like "is J (nij In) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point es timate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p( 7r) comprising prior information about 7r. From the prior p(7r) one can compute the posterior p(7rln), from which the distribution p(Iln) of the mutual information can be cal culated. We derive reliable and quickly computable approximations for p(Iln). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed. 1 Introduction The mutual information J (also called cross entropy) is a widely used information theoretic measure for the stochastic dependency of random variables [CT91, SooOO] . It is used, for instance, in learning Bayesian nets [Bun96, Hec98] , where stochasti cally dependent nodes shall be connected. The mutual information defined in (1) can be computed if the joint probabilities {7rij} of the two random variables z and J are known. -
Simulation of Moment, Cumulant, Kurtosis and the Characteristics Function of Dagum Distribution
264 IJCSNS International Journal of Computer Science and Network Security, VOL.17 No.8, August 2017 Simulation of Moment, Cumulant, Kurtosis and the Characteristics Function of Dagum Distribution Dian Kurniasari1*,Yucky Anggun Anggrainy1, Warsono1 , Warsito2 and Mustofa Usman1 1Department of Mathematics, Faculty of Mathematics and Sciences, Universitas Lampung, Indonesia 2Department of Physics, Faculty of Mathematics and Sciences, Universitas Lampung, Indonesia Abstract distribution is a special case of Generalized Beta II The Dagum distribution is a special case of Generalized Beta II distribution. Dagum (1977, 1980) distribution has been distribution with the parameter q=1, has 3 parameters, namely a used in studies of income and wage distribution as well as and p as shape parameters and b as a scale parameter. In this wealth distribution. In this context its features have been study some properties of the distribution: moment, cumulant, extensively analyzed by many authors, for excellent Kurtosis and the characteristics function of the Dagum distribution will be discussed. The moment and cumulant will be survey on the genesis and on empirical applications see [4]. discussed through Moment Generating Function. The His proposals enable the development of statistical characteristics function will be discussed through the expectation of the definition of characteristics function. Simulation studies distribution used to empirical income and wealth data that will be presented. Some behaviors of the distribution due to the could accommodate both heavy tails in empirical income variation values of the parameters are discussed. and wealth distributions, and also permit interior mode [5]. Key words: A random variable X is said to have a Dagum distribution Dagum distribution, moment, cumulant, kurtosis, characteristics function. -
Exponentiated Kumaraswamy-Dagum Distribution with Applications to Income and Lifetime Data Shujiao Huang and Broderick O Oluyede*
Huang and Oluyede Journal of Statistical Distributions and Applications 2014, 1:8 http://www.jsdajournal.com/content/1/1/8 RESEARCH Open Access Exponentiated Kumaraswamy-Dagum distribution with applications to income and lifetime data Shujiao Huang and Broderick O Oluyede* *Correspondence: [email protected] Abstract Department of Mathematical A new family of distributions called exponentiated Kumaraswamy-Dagum (EKD) Sciences, Georgia Southern University, Statesboro, GA 30458, distribution is proposed and studied. This family includes several well known USA sub-models, such as Dagum (D), Burr III (BIII), Fisk or Log-logistic (F or LLog), and new sub-models, namely, Kumaraswamy-Dagum (KD), Kumaraswamy-Burr III (KBIII), Kumaraswamy-Fisk or Kumaraswamy-Log-logistic (KF or KLLog), exponentiated Kumaraswamy-Burr III (EKBIII), and exponentiated Kumaraswamy-Fisk or exponentiated Kumaraswamy-Log-logistic (EKF or EKLLog) distributions. Statistical properties including series representation of the probability density function, hazard and reverse hazard functions, moments, mean and median deviations, reliability, Bonferroni and Lorenz curves, as well as entropy measures for this class of distributions and the sub-models are presented. Maximum likelihood estimates of the model parameters are obtained. Simulation studies are conducted. Examples and applications as well as comparisons of the EKD and its sub-distributions with other distributions are given. Mathematics Subject Classification (2000): 62E10; 62F30 Keywords: Dagum distribution; Exponentiated Kumaraswamy-Dagum distribution; Maximum likelihood estimation 1 Introduction Camilo Dagum proposed the distribution which is referred to as Dagum distribution in 1977. This proposal enable the development of statistical distributions used to fit empir- ical income and wealth data, that could accommodate heavy tails in income and wealth distributions. -
5. the Student T Distribution
Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way. -
Moments of the Product and Ratio of Two Correlated Chi-Square Variables
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Stat Papers (2009) 50:581–592 DOI 10.1007/s00362-007-0105-0 REGULAR ARTICLE Moments of the product and ratio of two correlated chi-square variables Anwar H. Joarder Received: 2 June 2006 / Revised: 8 October 2007 / Published online: 20 November 2007 © The Author(s) 2007 Abstract The exact probability density function of a bivariate chi-square distribu- tion with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such applica- tion is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution. Keywords Bivariate chi-square distribution · Moments · Product of correlated chi-square variables · Ratio of correlated chi-square variables Mathematics Subject Classification (2000) 62E15 · 60E05 · 60E10 1 Introduction Fisher (1915) derived the distribution of mean-centered sum of squares and sum of products in order to study the distribution of correlation coefficient from a bivariate nor- mal sample. Let X1, X2,...,X N (N > 2) be two-dimensional independent random vectors where X j = (X1 j , X2 j ) , j = 1, 2,...,N is distributed as a bivariate normal distribution denoted by N2(θ, ) with θ = (θ1,θ2) and a 2 × 2 covariance matrix = (σik), i = 1, 2; k = 1, 2. The sample mean-centered sums of squares and sum of products are given by a = N (X − X¯ )2 = mS2, m = N − 1,(i = 1, 2) ii j=1 ij i i = N ( − ¯ )( − ¯ ) = and a12 j=1 X1 j X1 X2 j X2 mRS1 S2, respectively. -
A Study of Non-Central Skew T Distributions and Their Applications in Data Analysis and Change Point Detection
A STUDY OF NON-CENTRAL SKEW T DISTRIBUTIONS AND THEIR APPLICATIONS IN DATA ANALYSIS AND CHANGE POINT DETECTION Abeer M. Hasan A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Arjun K. Gupta, Co-advisor Wei Ning, Advisor Mark Earley, Graduate Faculty Representative Junfeng Shang. Copyright c August 2013 Abeer M. Hasan All rights reserved iii ABSTRACT Arjun K. Gupta, Co-advisor Wei Ning, Advisor Over the past three decades there has been a growing interest in searching for distribution families that are suitable to analyze skewed data with excess kurtosis. The search started by numerous papers on the skew normal distribution. Multivariate t distributions started to catch attention shortly after the development of the multivariate skew normal distribution. Many researchers proposed alternative methods to generalize the univariate t distribution to the multivariate case. Recently, skew t distribution started to become popular in research. Skew t distributions provide more flexibility and better ability to accommodate long-tailed data than skew normal distributions. In this dissertation, a new non-central skew t distribution is studied and its theoretical properties are explored. Applications of the proposed non-central skew t distribution in data analysis and model comparisons are studied. An extension of our distribution to the multivariate case is presented and properties of the multivariate non-central skew t distri- bution are discussed. We also discuss the distribution of quadratic forms of the non-central skew t distribution. In the last chapter, the change point problem of the non-central skew t distribution is discussed under different settings. -
ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions
ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions F (x) = P r(X ≤ x) S(x) = 1 − F (x) dF (x) f(x) = dx H(x) = − ln S(x) dH(x) f(x) h(x) = = dx S(x) Functions of random variables Z 1 Expected Value E[g(x)] = g(x)f(x)dx −∞ 0 n n-th raw moment µn = E[X ] n n-th central moment µn = E[(X − µ) ] Variance σ2 = E[(X − µ)2] = E[X2] − µ2 µ µ0 − 3µ0 µ + 2µ3 Skewness γ = 3 = 3 2 1 σ3 σ3 µ µ0 − 4µ0 µ + 6µ0 µ2 − 3µ4 Kurtosis γ = 4 = 4 3 2 2 σ4 σ4 Moment generating function M(t) = E[etX ] Probability generating function P (z) = E[zX ] More concepts • Standard deviation (σ) is positive square root of variance • Coefficient of variation is CV = σ/µ • 100p-th percentile π is any point satisfying F (π−) ≤ p and F (π) ≥ p. If F is continuous, it is the unique point satisfying F (π) = p • Median is 50-th percentile; n-th quartile is 25n-th percentile • Mode is x which maximizes f(x) (n) n (n) • MX (0) = E[X ], where M is the n-th derivative (n) PX (0) • n! = P r(X = n) (n) • PX (1) is the n-th factorial moment of X. Bayes' Theorem P r(BjA)P r(A) P r(AjB) = P r(B) fY (yjx)fX (x) fX (xjy) = fY (y) Law of total probability 2 If Bi is a set of exhaustive (in other words, P r([iBi) = 1) and mutually exclusive (in other words P r(Bi \ Bj) = 0 for i 6= j) events, then for any event A, X X P r(A) = P r(A \ Bi) = P r(Bi)P r(AjBi) i i Correspondingly, for continuous distributions, Z P r(A) = P r(Ajx)f(x)dx Conditional Expectation Formula EX [X] = EY [EX [XjY ]] 3 Lesson 2: Parametric Distributions Forms of probability