DOCSLIB.ORG

Student's T-Distribution

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Student's T-Distribution Student's t-distribution From Wikipedia, the free encyclopedia Student's t In probability and statistics , Student's t- distribution (or simply the t-distribution ) is a Probability density function continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small. It plays a role in a number of widely-used statistical analyses, including the Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student's t- distribution also arises in the Bayesian analysis of data from a normal family. The t-distribution is symmetric and bell -shaped, like the normal distribution, but has heavier tails, Cumulative distribution function meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student's t- distribution is a special case of the generalised hyperbolic distribution . Contents ■ 1 Introduction ■ 1.1 History and etymology parameters: ν > 0 ■ 1.2 Examples degrees of freedom (real) support: ■ 2 Characterization ■ 2.1 Probability density function pdf: ■ 2.1.1 Derivation ■ 2.2 Cumulative distribution function cdf: ■ 3 Properties ■ 3.1 Moments ■ 3.2 Related distributions ■ 3.3 Monte Carlo sampling where 2F1 is the hypergeometric ■ 3.4 Integral of Student's function probability density function and p- mean: 0 for ν > 1, otherwise undefined value ■ 4 Related distributions median: 0 ■ 4.1 Three-parameter version ■ 4.2 Discrete version mode: 0 ■ 5 Special cases variance: ■ 5.1 ν = 1 , for ■ 5.2 ν = 2 , otherwise undefined ■ 6 Uses skewness: 0 for ν > 3 ■ 6.1 In frequentist statistical inference ex.kurtosis: ■ 6.1.1 Hypothesis testing ■ 6.1.2 Confidence intervals entropy: ■ 6.1.3 Prediction intervals ■ 6.2 Robust parametric modeling ■ 7 Table of selected values ■ 8 See also ■ ψ: digamma function, ■ 9 Notes ■ B: beta function ■ 10 References ■ 11 External links mgf: (Not defined) cf: Introduction [1] History and etymology ■ Kν(x): Bessel function In statistics, the t-distribution was first derived as a posterior distribution by Helmert and Lüroth.[2][3][4] In the English literature, a derivation of the t-distribution was published in 1908 by William Sealy Gosset [5] while he worked at the Guinness Brewery in Dublin. Since Gosset's employer forbade members of its staff from publishing scientific papers, his work was published under the pseudonym Student . The t-test and the associated theory became well-known through the work of R.A. Fisher, who called the distribution "Student's distribution". [6][7] Examples For examples of the use of this distribution, see Student's t test . Characterization Student's t-distribution is the probability distribution of the ratio [8] where ■ Z is normally distributed with expected value 0 and variance 1; ■ V has a chi-square distribution with ν ("nu") degrees of freedom; ■ Z and V are independent. While, for any given constant µ, is a random variable of noncentral t-distribution with noncentrality parameter µ. Probability density function Student's t-distribution has the probability density function where ν is the number of degrees of freedom and Γ is the Gamma function. For ν even, For ν odd, The overall shape of the probability density function of the t-distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean 0 and variance 1. The following images show the density of the t-distribution for increasing values of ν. The normal distribution is shown as a blue line for comparison. Note that the t-distribution (red line) becomes closer to the normal distribution as ν increases. Density of the t-distribution (red) for 1, 2, 3, 5, 10, and 30 df compared to normal distribution (blue). Previous plots shown in green. 1 degree of freedom 2 degrees of freedom 3 degrees of freedom 5 degrees of freedom 10 degrees of freedom 30 degrees of freedom Derivation Suppose X1, ..., Xn are independent values that are normally distributed with expected value µ and variance σ2. Let be the sample mean, and be the sample variance. It can be shown that the random variable has a chi -square distribution with n − 1 degrees of freedom (by Cochran's theorem ). It is readily shown that the quantity is normally distributed with mean 0 and variance 1, since the sample mean is normally distributed with mean μ and standard error . Moreover, it is possible to show that these two random variables—the normally distributed one and the chi-square-distributed one—are independent. Consequently the pivotal quantity , which differs from Z in that the exact standard deviation σ is replaced by the random variable Sn, has a Student's t-distribution as defined above. Notice that the unknown population variance σ2 does not appear in T, since it was in both the numerator and the denominators, so it canceled. Gosset's work showed that T has the probability density function with ν equal to n − 1. This may also be written as where B is the Beta function . The distribution of T is now called the t-distribution . The parameter ν is called the number of degrees of freedom . The distribution depends on ν, but not µ or σ; the lack of dependence on µ and σ is what makes the t-distribution important in both theory and practice. Gosset's result can be stated more generally. (See, for example, Hogg and Craig, Sections 4.4 and 4.8.) Let Z have a normal distribution with mean 0 and variance 1. Let V have a chi-square distribution with ν degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio has a t-distribution with ν degrees of freedom. Cumulative distribution function The cumulative distribution function is given by the regularized incomplete beta function , with Properties Moments The moments of the t -distribution are It should be noted that the term for 0 < k < ν, k even, may be simplified using the properties of the Gamma function to For a t-distribution with ν degrees of freedom, the expected value is 0, and its variance is ν/( ν − 2) if ν > 2. The skewness is 0 if ν > 3 and the excess kurtosis is 6/( ν − 4) if ν > 4. Related distributions ■ has a t-distribution if has a scaled inverse-χ2 distribution and has a normal distribution. ■ has an F-distribution if and has a Student's t- distribution. Monte Carlo sampling There are various approaches to constructing random samples from the Student distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function to uniform samples, e.g. in multi-dimensional applications basis on copula-dependency. In the case of stand-alone sampling, Bailey's 1994 extension of the Box-Muller method and its polar variation are easily deployed. It has the merit that it applies equally well to all real positive and negative degrees of freedom. Integral of Student's probability density function and p-value The function is the integral of Student's probability density function, ƒ(t) between −t and t. It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Therefore, the function can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of t and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly in t-tests. For the statistic t, with degrees of freedom, is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t > 0). It is defined for real t by the following formula: where B is the Beta function . For t > 0, there is a relation to the regularized incomplete beta function Ix (a, b) as follows: For statistical hypothesis testing this function is used to construct the p-value . Related distributions Three -parameter version Student's t distribution can be generalized to a three parameter location/scale family [9] that introduces a location parameter μ and an inverse scale parameter (i.e. precision) λ, and has a density defined by Other properties of this version of the distribution are [9]: This distribution results from compounding a Gaussian distribution with mean μ and unknown precision (the reciprocal of the variance), with a gamma distribution with parameters a = ν / 2 and b = ν / 2λ. In other words, the random variable X is assumed to have a normal distribution with an unknown precision distributed as gamma, and then this is marginalized over the gamma distribution. (The reason for the usefulness of this characterization is that the gamma distribution is the conjugate prior distribution of the precision of a Gaussian distribution. As a result, the three-parameter Student's t distribution arises naturally in many Bayesian inference problems.) The noncentral t -distribution is a different way of generalizing the t-distribution to include a location parameter.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Mathematical Statistics
    Mathematical Statistics MAS 713 Chapter 1.2 Previous subchapter 1 What is statistics ? 2 The statistical process 3 Population and samples 4 Random sampling Questions? Mathematical Statistics (MAS713) Ariel Neufeld 2 / 70 This subchapter Descriptive Statistics 1.2.1 Introduction 1.2.2 Types of variables 1.2.3 Graphical representations 1.2.4 Descriptive measures Mathematical Statistics (MAS713) Ariel Neufeld 3 / 70 1.2 Descriptive Statistics 1.2.1 Introduction Introduction As we said, statistics is the art of learning from data However, statistical data, obtained from surveys, experiments, or any series of measurements, are often so numerous that they are virtually useless unless they are condensed ; data should be presented in ways that facilitate their interpretation and subsequent analysis Naturally enough, the aspect of statistics which deals with organising, describing and summarising data is called descriptive statistics Mathematical Statistics (MAS713) Ariel Neufeld 4 / 70 1.2 Descriptive Statistics 1.2.2 Types of variables Types of variables Mathematical Statistics (MAS713) Ariel Neufeld 5 / 70 1.2 Descriptive Statistics 1.2.2 Types of variables Types of variables The range of available descriptive tools for a given data set depends on the type of the considered variable There are essentially two types of variables : 1 categorical (or qualitative) variables : take a value that is one of several possible categories (no numerical meaning) Ex. : gender, hair color, field of study, political affiliation, status, ... 2 numerical (or quantitative)
    [Show full text]
  • Concentration and Consistency Results for Canonical and Curved Exponential-Family Models of Random Graphs
    CONCENTRATION AND CONSISTENCY RESULTS FOR CANONICAL AND CURVED EXPONENTIAL-FAMILY MODELS OF RANDOM GRAPHS BY MICHAEL SCHWEINBERGER AND JONATHAN STEWART Rice University Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likeli- hood and M-estimators of a wide range of canonical and curved exponential- family models of random graphs with local dependence. All results are non- asymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an ap- plication, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes. 1. Introduction. Models of network data have witnessed a surge of interest in statistics and related areas [e.g., 31]. Such data arise in the study of, e.g., social networks, epidemics, insurgencies, and terrorist networks. Since the work of Holland and Leinhardt in the 1970s [e.g., 21], it is known that network data exhibit a wide range of dependencies induced by transitivity and other interesting network phenomena [e.g., 39]. Transitivity is a form of triadic closure in the sense that, when a node k is connected to two distinct nodes i and j, then i and j are likely to be connected as well, which suggests that edges are dependent [e.g., 39].
    [Show full text]
  • Applied Biostatistics Mean and Standard Deviation the Mean the Median Is Not the Only Measure of Central Value for a Distribution
    Health Sciences M.Sc. Programme Applied Biostatistics Mean and Standard Deviation The mean The median is not the only measure of central value for a distribution. Another is the arithmetic mean or average, usually referred to simply as the mean. This is found by taking the sum of the observations and dividing by their number. The mean is often denoted by a little bar over the symbol for the variable, e.g. x . The sample mean has much nicer mathematical properties than the median and is thus more useful for the comparison methods described later. The median is a very useful descriptive statistic, but not much used for other purposes. Median, mean and skewness The sum of the 57 FEV1s is 231.51 and hence the mean is 231.51/57 = 4.06. This is very close to the median, 4.1, so the median is within 1% of the mean. This is not so for the triglyceride data. The median triglyceride is 0.46 but the mean is 0.51, which is higher. The median is 10% away from the mean. If the distribution is symmetrical the sample mean and median will be about the same, but in a skew distribution they will not. If the distribution is skew to the right, as for serum triglyceride, the mean will be greater, if it is skew to the left the median will be greater. This is because the values in the tails affect the mean but not the median. Figure 1 shows the positions of the mean and median on the histogram of triglyceride.
    [Show full text]
  • Use of Statistical Tables
    TUTORIAL | SCOPE USE OF STATISTICAL TABLES Lucy Radford, Jenny V Freeman and Stephen J Walters introduce three important statistical distributions: the standard Normal, t and Chi-squared distributions PREVIOUS TUTORIALS HAVE LOOKED at hypothesis testing1 and basic statistical tests.2–4 As part of the process of statistical hypothesis testing, a test statistic is calculated and compared to a hypothesised critical value and this is used to obtain a P- value. This P-value is then used to decide whether the study results are statistically significant or not. It will explain how statistical tables are used to link test statistics to P-values. This tutorial introduces tables for three important statistical distributions (the TABLE 1. Extract from two-tailed standard Normal, t and Chi-squared standard Normal table. Values distributions) and explains how to use tabulated are P-values corresponding them with the help of some simple to particular cut-offs and are for z examples. values calculated to two decimal places. STANDARD NORMAL DISTRIBUTION TABLE 1 The Normal distribution is widely used in statistics and has been discussed in z 0.00 0.01 0.02 0.03 0.050.04 0.05 0.06 0.07 0.08 0.09 detail previously.5 As the mean of a Normally distributed variable can take 0.00 1.0000 0.9920 0.9840 0.9761 0.9681 0.9601 0.9522 0.9442 0.9362 0.9283 any value (−∞ to ∞) and the standard 0.10 0.9203 0.9124 0.9045 0.8966 0.8887 0.8808 0.8729 0.8650 0.8572 0.8493 deviation any positive value (0 to ∞), 0.20 0.8415 0.8337 0.8259 0.8181 0.8103 0.8206 0.7949 0.7872 0.7795 0.7718 there are an infinite number of possible 0.30 0.7642 0.7566 0.7490 0.7414 0.7339 0.7263 0.7188 0.7114 0.7039 0.6965 Normal distributions.
    [Show full text]
  • 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.
    [Show full text]
  • 1. How Different Is the T Distribution from the Normal?
    Statistics 101–106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M §7.1 and §7.2, ignoring starred parts. Reread M&M §3.2. The eects of estimated variances on normal approximations. t-distributions. Comparison of two means: pooling of estimates of variances, or paired observations. In Lecture 6, when discussing comparison of two Binomial proportions, I was content to estimate unknown variances when calculating statistics that were to be treated as approximately normally distributed. You might have worried about the effect of variability of the estimate. W. S. Gosset (“Student”) considered a similar problem in a very famous 1908 paper, where the role of Student’s t-distribution was first recognized. Gosset discovered that the effect of estimated variances could be described exactly in a simplified problem where n independent observations X1,...,Xn are taken from (, ) = ( + ...+ )/ a normal√ distribution, N . The sample mean, X X1 Xn n has a N(, / n) distribution. The random variable X Z = √ / n 2 2 Phas a standard normal distribution. If we estimate by the sample variance, s = ( )2/( ) i Xi X n 1 , then the resulting statistic, X T = √ s/ n no longer has a normal distribution. It has a t-distribution on n 1 degrees of freedom. Remark. I have written T , instead of the t used by M&M page 505. I find it causes confusion that t refers to both the name of the statistic and the name of its distribution. As you will soon see, the estimation of the variance has the effect of spreading out the distribution a little beyond what it would be if were used.
    [Show full text]
  • TR Multivariate Conditional Median Estimation
    Discussion Paper: 2004/02 TR multivariate conditional median estimation Jan G. de Gooijer and Ali Gannoun www.fee.uva.nl/ke/UvA-Econometrics Department of Quantitative Economics Faculty of Economics and Econometrics Universiteit van Amsterdam Roetersstraat 11 1018 WB AMSTERDAM The Netherlands TR Multivariate Conditional Median Estimation Jan G. De Gooijer1 and Ali Gannoun2 1 Department of Quantitative Economics University of Amsterdam Roetersstraat 11, 1018 WB Amsterdam, The Netherlands Telephone: +31—20—525 4244; Fax: +31—20—525 4349 e-mail: [email protected] 2 Laboratoire de Probabilit´es et Statistique Universit´e Montpellier II Place Eug`ene Bataillon, 34095 Montpellier C´edex 5, France Telephone: +33-4-67-14-3578; Fax: +33-4-67-14-4974 e-mail: [email protected] Abstract An affine equivariant version of the nonparametric spatial conditional median (SCM) is con- structed, using an adaptive transformation-retransformation (TR) procedure. The relative performance of SCM estimates, computed with and without applying the TR—procedure, are compared through simulation. Also included is the vector of coordinate conditional, kernel- based, medians (VCCMs). The methodology is illustrated via an empirical data set. It is shown that the TR—SCM estimator is more efficient than the SCM estimator, even when the amount of contamination in the data set is as high as 25%. The TR—VCCM- and VCCM estimators lack efficiency, and consequently should not be used in practice. Key Words: Spatial conditional median; kernel; retransformation; robust; transformation. 1 Introduction p s Let (X1, Y 1),...,(Xn, Y n) be independent replicates of a random vector (X, Y ) IR IR { } ∈ × where p > 1,s > 2, and n>p+ s.
    [Show full text]
  • This History of Modern Mathematical Statistics Retraces Their Development
    BOOK REVIEWS GORROOCHURN Prakash, 2016, Classic Topics on the History of Modern Mathematical Statistics: From Laplace to More Recent Times, Hoboken, NJ, John Wiley & Sons, Inc., 754 p. This history of modern mathematical statistics retraces their development from the “Laplacean revolution,” as the author so rightly calls it (though the beginnings are to be found in Bayes’ 1763 essay(1)), through the mid-twentieth century and Fisher’s major contribution. Up to the nineteenth century the book covers the same ground as Stigler’s history of statistics(2), though with notable differences (see infra). It then discusses developments through the first half of the twentieth century: Fisher’s synthesis but also the renewal of Bayesian methods, which implied a return to Laplace. Part I offers an in-depth, chronological account of Laplace’s approach to probability, with all the mathematical detail and deductions he drew from it. It begins with his first innovative articles and concludes with his philosophical synthesis showing that all fields of human knowledge are connected to the theory of probabilities. Here Gorrouchurn raises a problem that Stigler does not, that of induction (pp. 102-113), a notion that gives us a better understanding of probability according to Laplace. The term induction has two meanings, the first put forward by Bacon(3) in 1620, the second by Hume(4) in 1748. Gorroochurn discusses only the second. For Bacon, induction meant discovering the principles of a system by studying its properties through observation and experimentation. For Hume, induction was mere enumeration and could not lead to certainty. Laplace followed Bacon: “The surest method which can guide us in the search for truth, consists in rising by induction from phenomena to laws and from laws to forces”(5).
    [Show full text]
  • 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.
    [Show full text]
  • Estimation of Common Location and Scale Parameters in Nonregular Cases Ahmad Razmpour Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1982 Estimation of common location and scale parameters in nonregular cases Ahmad Razmpour Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Razmpour, Ahmad, "Estimation of common location and scale parameters in nonregular cases " (1982). Retrospective Theses and Dissertations. 7528. https://lib.dr.iastate.edu/rtd/7528 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed.
    [Show full text]
  • On a Problem Connected with Beta and Gamma Distributions by R
    ON A PROBLEM CONNECTED WITH BETA AND GAMMA DISTRIBUTIONS BY R. G. LAHA(i) 1. Introduction. The random variable X is said to have a Gamma distribution G(x;0,a)if du for x > 0, (1.1) P(X = x) = G(x;0,a) = JoT(a)" 0 for x ^ 0, where 0 > 0, a > 0. Let X and Y be two independently and identically distributed random variables each having a Gamma distribution of the form (1.1). Then it is well known [1, pp. 243-244], that the random variable W = X¡iX + Y) has a Beta distribution Biw ; a, a) given by 0 for w = 0, (1.2) PiW^w) = Biw;x,x)=\ ) u"-1il-u)'-1du for0<w<l, Ío T(a)r(a) 1 for w > 1. Now we can state the converse problem as follows : Let X and Y be two independently and identically distributed random variables having a common distribution function Fix). Suppose that W = Xj{X + Y) has a Beta distribution of the form (1.2). Then the question is whether £(x) is necessarily a Gamma distribution of the form (1.1). This problem was posed by Mauldon in [9]. He also showed that the converse problem is not true in general and constructed an example of a non-Gamma distribution with this property using the solution of an integral equation which was studied by Goodspeed in [2]. In the present paper we carry out a systematic investigation of this problem. In §2, we derive some general properties possessed by this class of distribution laws Fix).
    [Show full text]