Chapter 2 Random Variables and Distributions
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Chapter 6 Continuous Random Variables and Probability
EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2019 Chapter 6 Continuous Random Variables and Probability Distributions Chap 6-1 Probability Distributions Probability Distributions Ch. 5 Discrete Continuous Ch. 6 Probability Probability Distributions Distributions Binomial Uniform Hypergeometric Normal Poisson Exponential Chap 6-2/62 Continuous Probability Distributions § A continuous random variable is a variable that can assume any value in an interval § thickness of an item § time required to complete a task § temperature of a solution § height in inches § These can potentially take on any value, depending only on the ability to measure accurately. Chap 6-3/62 Cumulative Distribution Function § The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x F(x) = P(X £ x) § Let a and b be two possible values of X, with a < b. The probability that X lies between a and b is P(a < X < b) = F(b) -F(a) Chap 6-4/62 Probability Density Function The probability density function, f(x), of random variable X has the following properties: 1. f(x) > 0 for all values of x 2. The area under the probability density function f(x) over all values of the random variable X is equal to 1.0 3. The probability that X lies between two values is the area under the density function graph between the two values 4. The cumulative density function F(x0) is the area under the probability density function f(x) from the minimum x value up to x0 x0 f(x ) = f(x)dx 0 ò xm where -
5.1 Convergence in Distribution
556: MATHEMATICAL STATISTICS I CHAPTER 5: STOCHASTIC CONVERGENCE The following definitions are stated in terms of scalar random variables, but extend naturally to vector random variables defined on the same probability space with measure P . Forp example, some results 2 are stated in terms of the Euclidean distance in one dimension jXn − Xj = (Xn − X) , or for se- > quences of k-dimensional random variables Xn = (Xn1;:::;Xnk) , 0 1 1=2 Xk @ 2A kXn − Xk = (Xnj − Xj) : j=1 5.1 Convergence in Distribution Consider a sequence of random variables X1;X2;::: and a corresponding sequence of cdfs, FX1 ;FX2 ;::: ≤ so that for n = 1; 2; :: FXn (x) =P[Xn x] : Suppose that there exists a cdf, FX , such that for all x at which FX is continuous, lim FX (x) = FX (x): n−!1 n Then X1;:::;Xn converges in distribution to random variable X with cdf FX , denoted d Xn −! X and FX is the limiting distribution. Convergence of a sequence of mgfs or cfs also indicates conver- gence in distribution, that is, if for all t at which MX (t) is defined, if as n −! 1, we have −! () −!d MXi (t) MX (t) Xn X: Definition : DEGENERATE DISTRIBUTIONS The sequence of random variables X1;:::;Xn converges in distribution to constant c if the limiting d distribution of X1;:::;Xn is degenerate at c, that is, Xn −! X and P [X = c] = 1, so that { 0 x < c F (x) = X 1 x ≥ c Interpretation: A special case of convergence in distribution occurs when the limiting distribution is discrete, with the probability mass function only being non-zero at a single value, that is, if the limiting random variable is X, then P [X = c] = 1 and zero otherwise. -
Extremal Dependence Concepts
Extremal dependence concepts Giovanni Puccetti1 and Ruodu Wang2 1Department of Economics, Management and Quantitative Methods, University of Milano, 20122 Milano, Italy 2Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada Journal version published in Statistical Science, 2015, Vol. 30, No. 4, 485{517 Minor corrections made in May and June 2020 Abstract The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. -
Joint Probability Distributions
ST 380 Probability and Statistics for the Physical Sciences Joint Probability Distributions In many experiments, two or more random variables have values that are determined by the outcome of the experiment. For example, the binomial experiment is a sequence of trials, each of which results in success or failure. If ( 1 if the i th trial is a success Xi = 0 otherwise; then X1; X2;:::; Xn are all random variables defined on the whole experiment. 1 / 15 Joint Probability Distributions Introduction ST 380 Probability and Statistics for the Physical Sciences To calculate probabilities involving two random variables X and Y such as P(X > 0 and Y ≤ 0); we need the joint distribution of X and Y . The way we represent the joint distribution depends on whether the random variables are discrete or continuous. 2 / 15 Joint Probability Distributions Introduction ST 380 Probability and Statistics for the Physical Sciences Two Discrete Random Variables If X and Y are discrete, with ranges RX and RY , respectively, the joint probability mass function is p(x; y) = P(X = x and Y = y); x 2 RX ; y 2 RY : Then a probability like P(X > 0 and Y ≤ 0) is just X X p(x; y): x2RX :x>0 y2RY :y≤0 3 / 15 Joint Probability Distributions Two Discrete Random Variables ST 380 Probability and Statistics for the Physical Sciences Marginal Distribution To find the probability of an event defined only by X , we need the marginal pmf of X : X pX (x) = P(X = x) = p(x; y); x 2 RX : y2RY Similarly the marginal pmf of Y is X pY (y) = P(Y = y) = p(x; y); y 2 RY : x2RX 4 / 15 Joint -
Parameter Specification of the Beta Distribution and Its Dirichlet
%HWD'LVWULEXWLRQVDQG,WV$SSOLFDWLRQV 3DUDPHWHU6SHFLILFDWLRQRIWKH%HWD 'LVWULEXWLRQDQGLWV'LULFKOHW([WHQVLRQV 8WLOL]LQJ4XDQWLOHV -5HQpYDQ'RUSDQG7KRPDV$0D]]XFKL (Submitted January 2003, Revised March 2003) I. INTRODUCTION.................................................................................................... 1 II. SPECIFICATION OF PRIOR BETA PARAMETERS..............................................5 A. Basic Properties of the Beta Distribution...............................................................6 B. Solving for the Beta Prior Parameters...................................................................8 C. Design of a Numerical Procedure........................................................................12 III. SPECIFICATION OF PRIOR DIRICHLET PARAMETERS................................. 17 A. Basic Properties of the Dirichlet Distribution...................................................... 18 B. Solving for the Dirichlet prior parameters...........................................................20 IV. SPECIFICATION OF ORDERED DIRICHLET PARAMETERS...........................22 A. Properties of the Ordered Dirichlet Distribution................................................. 23 B. Solving for the Ordered Dirichlet Prior Parameters............................................ 25 C. Transforming the Ordered Dirichlet Distribution and Numerical Stability ......... 27 V. CONCLUSIONS........................................................................................................ 31 APPENDIX................................................................................................................... -
Package 'Distributional'
Package ‘distributional’ February 2, 2021 Title Vectorised Probability Distributions Version 0.2.2 Description Vectorised distribution objects with tools for manipulating, visualising, and using probability distributions. Designed to allow model prediction outputs to return distributions rather than their parameters, allowing users to directly interact with predictive distributions in a data-oriented workflow. In addition to providing generic replacements for p/d/q/r functions, other useful statistics can be computed including means, variances, intervals, and highest density regions. License GPL-3 Imports vctrs (>= 0.3.0), rlang (>= 0.4.5), generics, ellipsis, stats, numDeriv, ggplot2, scales, farver, digest, utils, lifecycle Suggests testthat (>= 2.1.0), covr, mvtnorm, actuar, ggdist RdMacros lifecycle URL https://pkg.mitchelloharawild.com/distributional/, https: //github.com/mitchelloharawild/distributional BugReports https://github.com/mitchelloharawild/distributional/issues Encoding UTF-8 Language en-GB LazyData true Roxygen list(markdown = TRUE, roclets=c('rd', 'collate', 'namespace')) RoxygenNote 7.1.1 1 2 R topics documented: R topics documented: autoplot.distribution . .3 cdf..............................................4 density.distribution . .4 dist_bernoulli . .5 dist_beta . .6 dist_binomial . .7 dist_burr . .8 dist_cauchy . .9 dist_chisq . 10 dist_degenerate . 11 dist_exponential . 12 dist_f . 13 dist_gamma . 14 dist_geometric . 16 dist_gumbel . 17 dist_hypergeometric . 18 dist_inflated . 20 dist_inverse_exponential . 20 dist_inverse_gamma -
Iam 530 Elements of Probability and Statistics
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 4-SOME DISCERETE AND CONTINUOUS DISTRIBUTION FUNCTIONS SOME DISCRETE PROBABILITY DISTRIBUTIONS Degenerate, Uniform, Bernoulli, Binomial, Poisson, Negative Binomial, Geometric, Hypergeometric DEGENERATE DISTRIBUTION • An rv X is degenerate at point k if 1, Xk P X x 0, ow. The cdf: 0, Xk F x P X x 1, Xk UNIFORM DISTRIBUTION • A finite number of equally spaced values are equally likely to be observed. 1 P(X x) ; x 1,2,..., N; N 1,2,... N • Example: throw a fair die. P(X=1)=…=P(X=6)=1/6 N 1 (N 1)(N 1) E(X) ; Var(X) 2 12 BERNOULLI DISTRIBUTION • An experiment consists of one trial. It can result in one of 2 outcomes: Success or Failure (or a characteristic being Present or Absent). • Probability of Success is p (0<p<1) 1 with probability p Xp;0 1 0 with probability 1 p P(X x) px (1 p)1 x for x 0,1; and 0 p 1 1 E(X ) xp(y) 0(1 p) 1p p y 0 E X 2 02 (1 p) 12 p p V (X ) E X 2 E(X ) 2 p p2 p(1 p) p(1 p) Binomial Experiment • Experiment consists of a series of n identical trials • Each trial can end in one of 2 outcomes: Success or Failure • Trials are independent (outcome of one has no bearing on outcomes of others) • Probability of Success, p, is constant for all trials • Random Variable X, is the number of Successes in the n trials is said to follow Binomial Distribution with parameters n and p • X can take on the values x=0,1,…,n • Notation: X~Bin(n,p) Consider outcomes of an experiment with 3 Trials : SSS y 3 P(SSS) P(Y 3) p(3) p3 SSF, SFS, FSS y 2 P(SSF SFS FSS) P(Y 2) p(2) 3p2 (1 p) SFF, FSF, FFS y 1 P(SFF FSF FFS ) P(Y 1) p(1) 3p(1 p)2 FFF y 0 P(FFF ) P(Y 0) p(0) (1 p)3 In General: n n! 1) # of ways of arranging x S s (and (n x) F s ) in a sequence of n positions x x!(n x)! 2) Probability of each arrangement of x S s (and (n x) F s ) p x (1 p)n x n 3)P(X x) p(x) p x (1 p)n x x 0,1,..., n x • Example: • There are black and white balls in a box. -
The Length-Biased Versus Random Sampling for the Binomial and Poisson Events Makarand V
Journal of Modern Applied Statistical Methods Volume 12 | Issue 1 Article 10 5-1-2013 The Length-Biased Versus Random Sampling for the Binomial and Poisson Events Makarand V. Ratnaparkhi Wright State University, [email protected] Uttara V. Naik-Nimbalkar Pune University, Pune, India Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Ratnaparkhi, Makarand V. and Naik-Nimbalkar, Uttara V. (2013) "The Length-Biased Versus Random Sampling for the Binomial and Poisson Events," Journal of Modern Applied Statistical Methods: Vol. 12 : Iss. 1 , Article 10. DOI: 10.22237/jmasm/1367381340 Available at: http://digitalcommons.wayne.edu/jmasm/vol12/iss1/10 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright © 2013 JMASM, Inc. May 2013, Vol. 12, No. 1, 54-57 1538 – 9472/13/$95.00 The Length-Biased Versus Random Sampling for the Binomial and Poisson Events Makarand V. Ratnaparkhi Uttara V. Naik-Nimbalkar Wright State University, Pune University, Dayton, OH Pune, India The equivalence between the length-biased and the random sampling on a non-negative, discrete random variable is established. The length-biased versions of the binomial and Poisson distributions are discussed. Key words: Length-biased data, weighted distributions, binomial, Poisson, convolutions. Introduction binomial distribution (probabilities), which was The occurrence of so-called length-biased data appropriate at that time. -
(Introduction to Probability at an Advanced Level) - All Lecture Notes
Fall 2018 Statistics 201A (Introduction to Probability at an advanced level) - All Lecture Notes Aditya Guntuboyina August 15, 2020 Contents 0.1 Sample spaces, Events, Probability.................................5 0.2 Conditional Probability and Independence.............................6 0.3 Random Variables..........................................7 1 Random Variables, Expectation and Variance8 1.1 Expectations of Random Variables.................................9 1.2 Variance................................................ 10 2 Independence of Random Variables 11 3 Common Distributions 11 3.1 Ber(p) Distribution......................................... 11 3.2 Bin(n; p) Distribution........................................ 11 3.3 Poisson Distribution......................................... 12 4 Covariance, Correlation and Regression 14 5 Correlation and Regression 16 6 Back to Common Distributions 16 6.1 Geometric Distribution........................................ 16 6.2 Negative Binomial Distribution................................... 17 7 Continuous Distributions 17 7.1 Normal or Gaussian Distribution.................................. 17 1 7.2 Uniform Distribution......................................... 18 7.3 The Exponential Density...................................... 18 7.4 The Gamma Density......................................... 18 8 Variable Transformations 19 9 Distribution Functions and the Quantile Transform 20 10 Joint Densities 22 11 Joint Densities under Transformations 23 11.1 Detour to Convolutions...................................... -
The Multivariate Normal Distribution=1See Last Slide For
Moment-generating Functions Definition Properties χ2 and t distributions The Multivariate Normal Distribution1 STA 302 Fall 2017 1See last slide for copyright information. 1 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Overview 1 Moment-generating Functions 2 Definition 3 Properties 4 χ2 and t distributions 2 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Joint moment-generating function Of a p-dimensional random vector x t0x Mx(t) = E e x t +x t +x t For example, M (t1; t2; t3) = E e 1 1 2 2 3 3 (x1;x2;x3) Just write M(t) if there is no ambiguity. Section 4.3 of Linear models in statistics has some material on moment-generating functions (optional). 3 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Uniqueness Proof omitted Joint moment-generating functions correspond uniquely to joint probability distributions. M(t) is a function of F (x). Step One: f(x) = @ ··· @ F (x). @x1 @xp @ @ R x2 R x1 For example, f(y1; y2) dy1dy2 @x1 @x2 −∞ −∞ 0 Step Two: M(t) = R ··· R et xf(x) dx Could write M(t) = g (F (x)). Uniqueness says the function g is one-to-one, so that F (x) = g−1 (M(t)). 4 / 40 Moment-generating Functions Definition Properties χ2 and t distributions g−1 (M(t)) = F (x) A two-variable example g−1 (M(t)) = F (x) −1 R 1 R 1 x1t1+x2t2 R x2 R x1 g −∞ −∞ e f(x1; x2) dx1dx2 = −∞ −∞ f(y1; y2) dy1dy2 5 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Theorem Two random vectors x1 and x2 are independent if and only if the moment-generating function of their joint distribution is the product of their moment-generating functions. -
Probability for Statistics: the Bare Minimum Prerequisites
Probability for Statistics: The Bare Minimum Prerequisites This note seeks to cover the very bare minimum of topics that students should be familiar with before starting the Probability for Statistics module. Most of you will have studied classical and/or axiomatic probability before, but given the various backgrounds it's worth being concrete on what students should know going in. Real Analysis Students should be familiar with the main ideas from real analysis, and at the very least be familiar with notions of • Limits of sequences of real numbers; • Limit Inferior (lim inf) and Limit Superior (lim sup) and their connection to limits; • Convergence of sequences of real-valued vectors; • Functions; injectivity and surjectivity; continuous and discontinuous functions; continuity from the left and right; Convex functions. • The notion of open and closed subsets of real numbers. For those wanting for a good self-study textbook for any of these topics I would recommend: Mathematical Analysis, by Malik and Arora. Complex Analysis We will need some tools from complex analysis when we study characteristic functions and subsequently the central limit theorem. Students should be familiar with the following concepts: • Imaginary numbers, complex numbers, complex conjugates and properties. • De Moivre's theorem. • Limits and convergence of sequences of complex numbers. • Fourier transforms and inverse Fourier transforms For those wanting a good self-study textbook for these topics I recommend: Introduction to Complex Analysis by H. Priestley. Classical Probability and Combinatorics Many students will have already attended a course in classical probability theory beforehand, and will be very familiar with these basic concepts. -
Using Learned Conditional Distributions As Edit Distance Jose Oncina, Marc Sebban
Using Learned Conditional Distributions as Edit Distance Jose Oncina, Marc Sebban To cite this version: Jose Oncina, Marc Sebban. Using Learned Conditional Distributions as Edit Distance. Structural, Syntactic, and Statistical Pattern Recognition, Joint IAPR International Workshops, SSPR 2006 and SPR 2006, Aug 2006, Hong Kong, China. pp 403-411, ISBN 3-540-37236-9. hal-00322429 HAL Id: hal-00322429 https://hal.archives-ouvertes.fr/hal-00322429 Submitted on 18 Sep 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Using Learned Conditional Distributions as Edit Distance⋆ Jose Oncina1 and Marc Sebban2 1 Dep. de Lenguajes y Sistemas Informaticos,´ Universidad de Alicante (Spain) [email protected] 2 EURISE, Universite´ de Saint-Etienne, (France) [email protected] Abstract. In order to achieve pattern recognition tasks, we aim at learning an unbiased stochastic edit distance, in the form of a finite-state transducer, from a corpus of (input,output) pairs of strings. Contrary to the state of the art methods, we learn a transducer independently on the marginal probability distribution of the input strings. Such an unbiased way to proceed requires to optimize the pa- rameters of a conditional transducer instead of a joint one.