DOCSLIB.ORG

A Note on Skew-Normal Distribution Approximation to the Negative Binomal Distribution

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Note on Skew-Normal Distribution Approximation to the Negative Binomal Distribution WSEAS TRANSACTIONS on MATHEMATICS Jyh-Jiuan Lin, Ching-Hui Chang, Rosemary Jou A NOTE ON SKEW-NORMAL DISTRIBUTION APPROXIMATION TO THE NEGATIVE BINOMAL DISTRIBUTION 1 2* 3 JYH-JIUAN LIN , CHING-HUI CHANG AND ROSEMARY JOU 1Department of Statistics Tamkang University 151 Ying-Chuan Road, Tamsui, Taipei County 251, Taiwan [email protected] 2Department of Applied Statistics and Information Science Ming Chuan University 5 De-Ming Road, Gui-Shan District, Taoyuan 333, Taiwan [email protected] 3 Graduate Institute of Management Sciences, Tamkang University 151 Ying-Chuan Road, Tamsui, Taipei County 251, Taiwan [email protected] Abstract: This article revisits the problem of approximating the negative binomial distribution. Its goal is to show that the skew-normal distribution, another alternative methodology, provides a much better approximation to negative binomial probabilities than the usual normal approximation. Key-Words: Central Limit Theorem, cumulative distribution function (cdf), skew-normal distribution, skew parameter. * Corresponding author ISSN: 1109-2769 32 Issue 1, Volume 9, January 2010 WSEAS TRANSACTIONS on MATHEMATICS Jyh-Jiuan Lin, Ching-Hui Chang, Rosemary Jou 1. INTRODUCTION where φ and Φ denote the standard normal pdf and cdf respectively. The This article revisits the problem of the parameter space is: μ ∈( −∞ , ∞), σ > 0 and negative binomial distribution approximation. Though the most commonly used solution of λ ∈(−∞ , ∞). Besides, the distribution approximating the negative binomial SN(0, 1, λ) with pdf distribution is by normal distribution, Peizer f( x | 0, 1, λ) = 2φ (x )Φ (λ x ) is called the and Pratt (1968) has investigated this problem standard skew-normal distribution. When thoroughly and proposed a different method λ = ,0 the above pdf (1.1) boils down to the which outperforms the normal approximation. N(μ , σ 2 ) pdf. Also parameter λ > 0 (< 0) However, the methodology used in Peizer and implies positively (negatively) skewed shape Pratt (1968) is rather complicated and difficult. of f( x | μ, σ , λ) above. For more With different motive, we propose another skew-normal distribution’s properties, one can alternative, the skew-normal distribution, and see Azzalini (1985), Gupta, Nguyen and show that it provides a much better Sanqui (2004), Arnold and Lin (2004) or approximation to negative binomial Chang et al. (2008). probabilities than the usual normal approximation. It seems that the skew-normal The skew-normal distribution is first distribution is a better choice than the normal introduced by O’Hagan and Leonard (1976), distribution to approximate the discrete which is a generalization of the normal distribution. The reason is that the distribution. Its definition is as follows. skew-normal distribution can take negatively skewed, symmetric or positively skewed A r.v. X is said to follow a skew-normal shape through its skew parameter and hence is distribution with location parameter μ, scale more versatile than the normal distribution. parameter σ and skew parameter λ (the Comparing to the normal distribution, Chang SN(μ , σ 2 , λ) distribution) if the pdf of X is et al. (2008) has shown that the extra (skew) given as parameter λ would provide a better approximation of the binomial distribution by f( x |μ , σ , λ) the SN(μ, σ2 , λ) distribution. Besides, = (2 /σ )φ ((x −μ ) /σ ) Φ (λ (x − μ ) /σ ), (1.1) Chang et al. (2008) has also mentioned that it ISSN: 1109-2769 33 Issue 1, Volume 9, January 2010 WSEAS TRANSACTIONS on MATHEMATICS Jyh-Jiuan Lin, Ching-Hui Chang, Rosemary Jou should also work for negative binomial applying to various areas. For the applications distribution approximation. However, a lot of of the CLT, please see Wang and Wei (2008), details of the numerical results are not Giribon, Revetria and Oliva (2007), Takacs, reported in Chang et al. (2008). Therefore, we Nagy and Vujacic (2007) and Mceachen et al will show that the extra (skew) parameter λ (2005). By the Central Limit Theorem, the cdf would provide a better approximation of the of NB(r, p) (negative binomial distribution negative binomial distribution by the where Y represents the number of failures th SN(μ , σ2 , λ) distribution than the normal before the r success) at k, k ∈ ,2,1,0{ K}, distribution in this paper along with the Fr, p ( k ), i.e., P( Y≤ k), is approximated by details. the normal cdf as The skew-normal distribution and its Fr, p ()k ≈ Φ((p ( k + 0.5)− rq ) / rq ) usefulness to approximate the negative binomial distribution is discussed in the next where q = 1− p. section. The usual normal approximation method is also introduced in Section 2 and 2.2. Skew-Normal Approximation numerical results and comparisons are For a given NB(r, p) distribution, the provided in Section 3. parameters of the approximating (or, matching) SN(μ , σ2 , λ) distribution are 2. APPROXIMATION BY THE found by equating the first three central NORMAL AND SKEW-NORMAL moments, i.e., DISTRIBUTIONS (a) EY() = rq /(p= E X ) In this section we’ll see how the 2 SN(μ , σ2 , λ) distribution can be used to = μ + σ ( 2 /π )( λ / 1+ λ ); approximate a negative binomial distribution. (b) EY( −E( Y ))2 = rq / p2 = E ( X − E ( X ))2 2.1. Normal Approximation by the CLT = σ 2{1− (2 /π ) λ2 /(1+ λ2 )}; It is a common practice to approximate the sampling distribution of the sample mean by (c ) EY( −E( Y ))3 = rq (1 + q ) / p3 the fact of the Central Limit Theorem (CLT). = E(XEX− ( ))3 The CLT is a very powerful theorem and is = σ 3 ( 2 /π )( λ / 1+ λ2 ) 3 ((4 / π )− 1). ISSN: 1109-2769 34 Issue 1, Volume 9, January 2010 WSEAS TRANSACTIONS on MATHEMATICS Jyh-Jiuan Lin, Ching-Hui Chang, Rosemary Jou Solving the above three equations is in order to make the skew-normal straight forward. We obtain the values of λ, approximation work, i.e., σ and μ as follows. Fr, p ( k )≈ Ψλ (( k + 0.5− μ ) /σ ). Since the skewness of the skew-normal distribution lies (A) Given (r , p), the above last two in (-0.99527, 0.99527), the skew-nor mal equations yield approximation is applied only when the λ = {(rq (2 /π )((4 / π )− 1) /(1 + q ))2 / 3 skewness of NB(r, p), (2−p ) / r (1 − p ), lies in this interval. However, the negative +(2 /π ) − 1}−1/ 2 binomial distribution is always positively (B) Given (r , p) and after obtaining λ as skewed. Hence, strictly speaking, the above, σ is found as restriction is that the skewness of NB(r, p) lies in (0, 0.99527). σ = (rq / p ) /{1− (2/π ) λ2 /(1+ λ2 )} 1/2 . 2.4. Evaluation Criterion (C) Finally, after obtaining λ and σ as Chang et. al. (2008) uses the “maximal above, get μ as absolute error (MABS)” of the skew-normal approximation to investigate the binomial μ =( rq/) p −σ (2/)/1 π λ+ λ2 . approximation B(n, p) as in Schader and Schmid (1989) (though their work is related After obtaining λ, σ and μ as to the normal approximation to the binomial given above, the negative binomial cdf is distribution). MABS can be substantial when p approximated as is near 0 or 1. In this paper we will also adopt the MABS criterion for this negative binomial F( k )≈ k+0.5 f ( x |μ , σ, λ)dx r, p ∫−∞ approximation problem and it is defined as = Ψλ ((k + 0.5− μ ) /σ ), * c MABS( r , p)= max | Fr, p ( k )− Fr, p ( k ) | k∈ RY where Ψλ (c ) = 2φ (z )Φ ( λ z ) dz is the ∫−∞ (2.1) standard skew-normal cdf. * where Fr, p ( k )= Φ (( p ( k + 0.5) − rq ) / rq ) 2.3. Restriction is for the normal case and A mild restriction is put on r and p, which is * Fr, p () k= Ψλ (( k + 0.5 − μ ) / σ ) is for the (2−p ) / r (1 − p ) < 0.99527, ISSN: 1109-2769 35 Issue 1, Volume 9, January 2010 WSEAS TRANSACTIONS on MATHEMATICS Jyh-Jiuan Lin, Ching-Hui Chang, Rosemary Jou skew-normal case. Y~ N ( r (1− p )/ p = 6.6667, r (1− p )/ p2 = 8.8889) (3.2) 3. EXAMPLES AND NUMERICAL RESULTS ENA= Fr, p ( k ) − Φ (( p ( k + 0.5) − rq ) / rq ) As a demonstration of the improved (3.3) approximation of the negative binomial distribution by the skew-normal distribution, ESA= Fr, p ( k )− Ψλ (( k + 0.5 − μ ) /σ ) (3.4) we first show the matching skew-normal (as Next, we calculate the MABS (see Eq. in Eq. (3.1) and normal (as in Eq. (3.2)) pdfs (2.1)) for the skew-normal approximations as for the negative binomial distribution well as the normal approximations ()Y~ NB ( r= 20, p = 0.75 ) in the Figure 3.1 (henceforth called MABS(SN) and MABS(N), and the errors are reported in the Table 3.1. respectively) for various r and p. The results Here ENA and ESA stand for “Error in are reported in Table 3.2. On the other hand, Normal Approximation” and “Error in Figure 3.2 shows the plots of MABS(SN) and Skew-Normal Approximation”, and they are Figure 3.3 shows the plots of MABS(N) as a defined as in (3.3) and (3.4) respectively. It function of p (for fixed r = 20) and r (for can be seen that the skew-normal curve fixed p = 0.75). Plots for other values of r follows the asymmetric negative binomial showed almost identical patterns. The results distribution closely. and plots both clearly show that the Y~ SN (μ = 3.4107, σ 2 = (4.4148)2 ,λ = 2.4224) skew-normal approximation is much superior (3.1) to the usual normal approximation.
Load more

Recommended publications
  • The Beta-Binomial Distribution Introduction Bayesian Derivation
    In Danish: 2005-09-19 / SLB Translated: 2008-05-28 / SLB Bayesian Statistics, Simulation and Software The beta-binomial distribution I have translated this document, written for another course in Danish, almost as is. I have kept the references to Lee, the textbook used for that course. Introduction In Lee: Bayesian Statistics, the beta-binomial distribution is very shortly mentioned as the predictive distribution for the binomial distribution, given the conjugate prior distribution, the beta distribution. (In Lee, see pp. 78, 214, 156.) Here we shall treat it slightly more in depth, partly because it emerges in the WinBUGS example in Lee x 9.7, and partly because it possibly can be useful for your project work. Bayesian Derivation We make n independent Bernoulli trials (0-1 trials) with probability parameter π. It is well known that the number of successes x has the binomial distribution. Considering the probability parameter π unknown (but of course the sample-size parameter n is known), we have xjπ ∼ bin(n; π); where in generic1 notation n p(xjπ) = πx(1 − π)1−x; x = 0; 1; : : : ; n: x We assume as prior distribution for π a beta distribution, i.e. π ∼ beta(α; β); 1By this we mean that p is not a fixed function, but denotes the density function (in the discrete case also called the probability function) for the random variable the value of which is argument for p. 1 with density function 1 p(π) = πα−1(1 − π)β−1; 0 < π < 1: B(α; β) I remind you that the beta function can be expressed by the gamma function: Γ(α)Γ(β) B(α; β) = : (1) Γ(α + β) In Lee, x 3.1 is shown that the posterior distribution is a beta distribution as well, πjx ∼ beta(α + x; β + n − x): (Because of this result we say that the beta distribution is conjugate distribution to the binomial distribution.) We shall now derive the predictive distribution, that is finding p(x).
    [Show full text]
  • 9. Binomial Distribution
    J. K. SHAH CLASSES Binomial Distribution 9. Binomial Distribution THEORETICAL DISTRIBUTION (Exists only in theory) 1. Theoretical Distribution is a distribution where the variables are distributed according to some definite mathematical laws. 2. In other words, Theoretical Distributions are mathematical models; where the frequencies / probabilities are calculated by mathematical computation. 3. Theoretical Distribution are also called as Expected Variance Distribution or Frequency Distribution 4. THEORETICAL DISTRIBUTION DISCRETE CONTINUOUS BINOMIAL POISSON Distribution Distribution NORMAL OR Student’s ‘t’ Chi-square Snedecor’s GAUSSIAN Distribution Distribution F-Distribution Distribution A. Binomial Distribution (Bernoulli Distribution) 1. This distribution is a discrete probability Distribution where the variable ‘x’ can assume only discrete values i.e. x = 0, 1, 2, 3,....... n 2. This distribution is derived from a special type of random experiment known as Bernoulli Experiment or Bernoulli Trials , which has the following characteristics (i) Each trial must be associated with two mutually exclusive & exhaustive outcomes – SUCCESS and FAILURE . Usually the probability of success is denoted by ‘p’ and that of the failure by ‘q’ where q = 1-p and therefore p + q = 1. (ii) The trials must be independent under identical conditions. (iii) The number of trial must be finite (countably finite). (iv) Probability of success and failure remains unchanged throughout the process. : 442 : J. K. SHAH CLASSES Binomial Distribution Note 1 : A ‘trial’ is an attempt to produce outcomes which is neither sure nor impossible in nature. Note 2 : The conditions mentioned may also be treated as the conditions for Binomial Distributions. 3. Characteristics or Properties of Binomial Distribution (i) It is a bi parametric distribution i.e.
    [Show full text]
  • 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
    [Show full text]
  • 3.2.3 Binomial Distribution
    3.2.3 Binomial Distribution The binomial distribution is based on the idea of a Bernoulli trial. A Bernoulli trail is an experiment with two, and only two, possible outcomes. A random variable X has a Bernoulli(p) distribution if 8 > <1 with probability p X = > :0 with probability 1 − p, where 0 ≤ p ≤ 1. The value X = 1 is often termed a “success” and X = 0 is termed a “failure”. The mean and variance of a Bernoulli(p) random variable are easily seen to be EX = (1)(p) + (0)(1 − p) = p and VarX = (1 − p)2p + (0 − p)2(1 − p) = p(1 − p). In a sequence of n identical, independent Bernoulli trials, each with success probability p, define the random variables X1,...,Xn by 8 > <1 with probability p X = i > :0 with probability 1 − p. The random variable Xn Y = Xi i=1 has the binomial distribution and it the number of sucesses among n independent trials. The probability mass function of Y is µ ¶ ¡ ¢ n ¡ ¢ P Y = y = py 1 − p n−y. y For this distribution, t n EX = np, Var(X) = np(1 − p),MX (t) = [pe + (1 − p)] . 1 Theorem 3.2.2 (Binomial theorem) For any real numbers x and y and integer n ≥ 0, µ ¶ Xn n (x + y)n = xiyn−i. i i=0 If we take x = p and y = 1 − p, we get µ ¶ Xn n 1 = (p + (1 − p))n = pi(1 − p)n−i. i i=0 Example 3.2.2 (Dice probabilities) Suppose we are interested in finding the probability of obtaining at least one 6 in four rolls of a fair die.
    [Show full text]
  • Solutions to the Exercises
    Solutions to the Exercises Chapter 1 Solution 1.1 (a) Your computer may be programmed to allocate borderline cases to the next group down, or the next group up; and it may or may not manage to follow this rule consistently, depending on its handling of the numbers involved. Following a rule which says 'move borderline cases to the next group up', these are the five classifications. (i) 1.0-1.2 1.2-1.4 1.4-1.6 1.6-1.8 1.8-2.0 2.0-2.2 2.2-2.4 6 6 4 8 4 3 4 2.4-2.6 2.6-2.8 2.8-3.0 3.0-3.2 3.2-3.4 3.4-3.6 3.6-3.8 6 3 2 2 0 1 1 (ii) 1.0-1.3 1.3-1.6 1.6-1.9 1.9-2.2 2.2-2.5 10 6 10 5 6 2.5-2.8 2.8-3.1 3.1-3.4 3.4-3.7 7 3 1 2 (iii) 0.8-1.1 1.1-1.4 1.4-1.7 1.7-2.0 2.0-2.3 2 10 6 10 7 2.3-2.6 2.6-2.9 2.9-3.2 3.2-3.5 3.5-3.8 6 4 3 1 1 (iv) 0.85-1.15 1.15-1.45 1.45-1.75 1.75-2.05 2.05-2.35 4 9 8 9 5 2.35-2.65 2.65-2.95 2.95-3.25 3.25-3.55 3.55-3.85 7 3 3 1 1 (V) 0.9-1.2 1.2-1.5 1.5-1.8 1.8-2.1 2.1-2.4 6 7 11 7 4 2.4-2.7 2.7-3.0 3.0-3.3 3.3-3.6 3.6-3.9 7 4 2 1 1 (b) Computer graphics: the diagrams are shown in Figures 1.9 to 1.11.
    [Show full text]
  • Chapter 5 Sections
    Chapter 5 Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions Continuous univariate distributions: 5.6 Normal distributions 5.7 Gamma distributions Just skim 5.8 Beta distributions Multivariate distributions Just skim 5.9 Multinomial distributions 5.10 Bivariate normal distributions 1 / 43 Chapter 5 5.1 Introduction Families of distributions How: Parameter and Parameter space pf /pdf and cdf - new notation: f (xj parameters ) Mean, variance and the m.g.f. (t) Features, connections to other distributions, approximation Reasoning behind a distribution Why: Natural justification for certain experiments A model for the uncertainty in an experiment All models are wrong, but some are useful – George Box 2 / 43 Chapter 5 5.2 Bernoulli and Binomial distributions Bernoulli distributions Def: Bernoulli distributions – Bernoulli(p) A r.v. X has the Bernoulli distribution with parameter p if P(X = 1) = p and P(X = 0) = 1 − p. The pf of X is px (1 − p)1−x for x = 0; 1 f (xjp) = 0 otherwise Parameter space: p 2 [0; 1] In an experiment with only two possible outcomes, “success” and “failure”, let X = number successes. Then X ∼ Bernoulli(p) where p is the probability of success. E(X) = p, Var(X) = p(1 − p) and (t) = E(etX ) = pet + (1 − p) 8 < 0 for x < 0 The cdf is F(xjp) = 1 − p for 0 ≤ x < 1 : 1 for x ≥ 1 3 / 43 Chapter 5 5.2 Bernoulli and Binomial distributions Binomial distributions Def: Binomial distributions – Binomial(n; p) A r.v.
    [Show full text]
  • 3 Multiple Discrete Random Variables
    3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F, P) and now we have two discrete random variables X and Y on it. They have probability mass functions fX (x) and fY (y). However, knowing these two functions is not enough. We illustrate this with an example. Example: We flip a fair coin twice and define several RV’s: Let X1 be the number of heads on the first flip. (So X1 = 0, 1.) Let X2 be the number of heads on the second flip. Let Y = 1 − X1. All three RV’s have the same p.m.f. : f(0) = 1/2, f(1) = 1/2. So if we only look at p.m.f.’s of individual RV’s, the pair X1,X2 looks just like the pair X1,Y . But these pairs are very different. For example, with the pair X1,Y , if we know the value of X1 then the value of Y is determined. But for the pair X1,X2, knowning the value of X1 tells us nothing about X2. Later we will define “independence” for RV’s and see that X1,X2 are an example of a pair of independent RV’s. We need to encode information from P about what our random variables do together, so we introduce the following object. Definition 1. If X and Y are discrete RV’s, their joint probability mass function is f(x, y)= P(X = x, Y = y) As always, the comma in the event X = x, Y = y means “and”.
    [Show full text]
  • Discrete Distributions: Empirical, Bernoulli, Binomial, Poisson
    Empirical Distributions An empirical distribution is one for which each possible event is assigned a probability derived from experimental observation. It is assumed that the events are independent and the sum of the probabilities is 1. An empirical distribution may represent either a continuous or a discrete distribution. If it represents a discrete distribution, then sampling is done “on step”. If it represents a continuous distribution, then sampling is done via “interpolation”. The way the table is described usually determines if an empirical distribution is to be handled discretely or continuously; e.g., discrete description continuous description value probability value probability 10 .1 0 – 10- .1 20 .15 10 – 20- .15 35 .4 20 – 35- .4 40 .3 35 – 40- .3 60 .05 40 – 60- .05 To use linear interpolation for continuous sampling, the discrete points on the end of each step need to be connected by line segments. This is represented in the graph below by the green line segments. The steps are represented in blue: rsample 60 50 40 30 20 10 0 x 0 .5 1 In the discrete case, sampling on step is accomplished by accumulating probabilities from the original table; e.g., for x = 0.4, accumulate probabilities until the cumulative probability exceeds 0.4; rsample is the event value at the point this happens (i.e., the cumulative probability 0.1+0.15+0.4 is the first to exceed 0.4, so the rsample value is 35). In the continuous case, the end points of the probability accumulation are needed, in this case x=0.25 and x=0.65 which represent the points (.25,20) and (.65,35) on the graph.
    [Show full text]
  • Hand-Book on STATISTICAL DISTRIBUTIONS for Experimentalists
    Internal Report SUF–PFY/96–01 Stockholm, 11 December 1996 1st revision, 31 October 1998 last modification 10 September 2007 Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists by Christian Walck Particle Physics Group Fysikum University of Stockholm (e-mail: [email protected]) Contents 1 Introduction 1 1.1 Random Number Generation .............................. 1 2 Probability Density Functions 3 2.1 Introduction ........................................ 3 2.2 Moments ......................................... 3 2.2.1 Errors of Moments ................................ 4 2.3 Characteristic Function ................................. 4 2.4 Probability Generating Function ............................ 5 2.5 Cumulants ......................................... 6 2.6 Random Number Generation .............................. 7 2.6.1 Cumulative Technique .............................. 7 2.6.2 Accept-Reject technique ............................. 7 2.6.3 Composition Techniques ............................. 8 2.7 Multivariate Distributions ................................ 9 2.7.1 Multivariate Moments .............................. 9 2.7.2 Errors of Bivariate Moments .......................... 9 2.7.3 Joint Characteristic Function .......................... 10 2.7.4 Random Number Generation .......................... 11 3 Bernoulli Distribution 12 3.1 Introduction ........................................ 12 3.2 Relation to Other Distributions ............................. 12 4 Beta distribution 13 4.1 Introduction .......................................
    [Show full text]
  • Statistical Distributions
    CHAPTER 10 Statistical Distributions In this chapter, we shall present some probability distributions that play a central role in econometric theory. First, we shall present the distributions of some discrete random variables that have either a finite set of values or that take values that can be indexed by the entire set of positive integers. We shall also present the multivariate generalisations of one of these distributions. Next, we shall present the distributions of some continuous random variables that take values in intervals of the real line or over the entirety of the real line. Amongst these is the normal distribution, which is of prime importance and for which we shall consider, in detail, the multivariate extensions. Associated with the multivariate normal distribution are the so-called sam- pling distributions that are important in the theory of statistical inference. We shall consider these distributions in the final section of the chapter, where it will transpire that they are special cases of the univariate distributions described in the preceding section. Discrete Distributions Suppose that there is a population of N elements, Np of which belong to class and N(1 p) to class c. When we select n elements at random from the populationA in n −successive trials,A we wish to know the probability of the event that x of them will be in and that n x of them will be in c. The probability will Abe affected b−y the way in which theA n elements are selected; and there are two ways of doing this. Either they can be put aside after they have been sampled, or else they can be restored to the population.
    [Show full text]
  • Some Continuous and Discrete Distributions Table of Contents I
    Some continuous and discrete distributions Table of contents I. Continuous distributions and transformation rules. A. Standard uniform distribution U[0; 1]. B. Uniform distribution U[a; b]. C. Standard normal distribution N(0; 1). D. Normal distribution N(¹; ). E. Standard exponential distribution. F. Exponential distribution with mean ¸. G. Standard Gamma distribution ¡(r; 1). H. Gamma distribution ¡(r; ¸). II. Discrete distributions and transformation rules. A. Bernoulli random variables. B. Binomial distribution. C. Poisson distribution. D. Geometric distribution. E. Negative binomial distribution. F. Hypergeometric distribution. 1 Continuous distributions. Each continuous distribution has a \standard" version and a more general rescaled version. The transformation from one to the other is always of the form Y = aX + b, with a > 0, and the resulting identities: y b fX ¡ f (y) = a (1) Y ³a ´ y b FY (y) = FX ¡ (2) Ã a ! E(Y ) = aE(X) + b (3) V ar(Y ) = a2V ar(X) (4) bt MY (t) = e MX (at) (5) 1 1.1 Standard uniform U[0; 1] This distribution is \pick a random number between 0 and 1". 1 if 0 < x < 1 fX (x) = ½ 0 otherwise 0 if x 0 · FX (x) = x if 0 x 1 8 · · < 1 if x 1 ¸ E(X) = 1:=2 V ar(X) = 1=12 et 1 M (t) = ¡ X t 1.2 Uniform U[a; b] This distribution is \pick a random number between a and b". To get a random number between a and b, take a random number between 0 and 1, multiply it by b a, and add a. The properties of this random variable are obtained by applying¡ rules (1{5) to the previous subsection.
    [Show full text]
  • Discrete Random Variables and Probability Distributions
    Discrete Random 3 Variables and Probability Distributions Stat 4570/5570 Based on Devore’s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer to the outcomes of such experiments as a “random variable”. Why is it called a “random variable”? 2 Random Variables Definition For a given sample space S of some experiment, a random variable (r.v.) is a rule that associates a number with each outcome in the sample space S. In mathematical language, a random variable is a “function” whose domain is the sample space and whose range is the set of real numbers: X : R S ! So, for any event s, we have X(s)=x is a real number. 3 Random Variables Notation! 1. Random variables - usually denoted by uppercase letters near the end of our alphabet (e.g. X, Y). 2. Particular value - now use lowercase letters, such as x, which correspond to the r.v. X. Birth weight example 4 Two Types of Random Variables A discrete random variable: Values constitute a finite or countably infinite set A continuous random variable: 1. Its set of possible values is the set of real numbers R, one interval, or a disjoint union of intervals on the real line (e.g., [0, 10] ∪ [20, 30]). 2. No one single value of the variable has positive probability, that is, P(X = c) = 0 for any possible value c. Only intervals have positive probabilities. 5 Probability Distributions for Discrete Random Variables Probabilities assigned to various outcomes in the sample space S, in turn, determine probabilities associated with the values of any particular random variable defined on S.
    [Show full text]