DOCSLIB.ORG

Chapter 5 the Normal Distribution

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 5 the Normal Distribution Chapter 5 The normal distribution This chapter deals in detail with one of the most versatile models for variation, the normal distribution or 'bell-shaped' curve. You will learn how to use printed tables to calculate normal probabilities. The normal curve also provides a useful approxi- mation to other probability distributions: this is one of the consequences of the central limit theorem. In Chapter 2, Section 2.4 you were introduced to an important continuous distribution called the normal distribution. It was noted that many real data sets can reasonably be treated as though they were a random sample from the normal distribution and it was remarked that the normal distribution turns out to play a central role in statistical theory as well as in practice. This entire chapter is devoted to the study of the normal distribution. The chapter begins with a review of all that has been said so far about the normal distribution. The main point to bear in mind is that in many cases a probability model for random variation follows necessarily as a mathematical consequence of certain assumptions: for instance, many random processes can be modelled as sets or sequences of Bernoulli trials, the distribution theory following from the twin assumptions that the trials are independent and that the probability of success from trial to trial remains constant. Quite often, however, data arise from a situation for which no model has been proposed: nevertheless, even when the data sets arise from entirely different sampling contexts, they often seem to acquire a characteristic peaked and symmetric shape that is essentially the same. This shape may often be adequately rep- resented through a normal model. The review is followed by an account of the genesis of the normal distribution. In Section 5.2, you will discover how to calculate normal probabilities. As for any other continuous probability distribution, probabilities are found by calculating areas under the curve of the probability density function. But for the normal distribution, this is not quite straightforward, because applying the technique of integration does not in this case lead to a formula that is easy to write down. So, in practice, probabilities are found by referring to printed tables, or by using a computer. The remaining sections of the chapter deal with one of the fundamental the- orems in statistics and with some of the consequences of it. It is called the central limit theorem. This is a theorem due to Pierre Simon Laplace (1749- 1827) that was read before the Academy of Sciences in Paris on 9 April 1810. The theorem is a major mathematical statement: however, we shall be con- cerned not with the details of its proof, but with its application to statistical problems. Elements of Statistics 5.1 Some history 5.1.1 Review The review begins with a set of data collected a long time ago. During the mapping of the state of Massachusetts in America, one hundred readings were taken on the error involved when measuring angles. The error was measured in minutes (a minute is 1/60 of a degree). The data are shown in Table 5.1. Table 5.1 Errors in angular measurements United States Coast Survey Report (1854). The error was calculated Error (in minutes) Frequency by subtracting each measurement Between +6 and +5 1 from 'the most probable' value. 2 Between +5 and +4 Frequency Between +4 and +3 2 Between +3 and +2 3 Between +2 and +l 13 Between +l and 0 26 Between 0 and -1 26 Between -1 and -2 17 Between -2 and -3 8 Between -3 and -4 2 10 A histogram of this sample is given in Figure 5.1. This graphical represen- tation shows clearly the main characteristics of the data: the histogram is unimodal (it possesses just one mode) and it is roughly symmetric about that mode. -4.0 -2.0 0.0 2.0 4.0 6.0 Error (minutes) Another histogram, which corresponds to a different data set, is shown in Figure 5.1 Errors in angular Figure 5.2. You have seen these data before. measurements (minutes) This is a graphical representation of the sample of Scottish soldiers' chest Frequency measurements that you met in Chapter 2, Section 2.4. This histogram is also unimodal and roughly symmetric. The common characteristics of the shape of both the histograms in Figures 5.1 and 5.2 are shared with the normal distribution whose p.d.f. is illustrated in Figure 5.3. 800 34 36 38 40 42 44 46 48 Chest (inches) Figure 5.2 Chest measurements of Scottish soldiers (inches) Figure 5.9 The normal p.d.f. For clarity, the vertical axis has been omitted in this graph of the normal density function. What is it about Figures 5.1, 5.2 and 5.3 that makes them appear similar? Well, each diagram starts at a low level on the left-hand side, rises steadily Chapter 5 Section 5.1 until reaching a maximum in the centre and then decreases, at the same rate that it increased, to a low value towards the right-hand side. The diagrams are unimodal and symmetric about their modes (although this symmetry is only approximate for the two data sets). A single descriptive word often used to describe the shape of the normal p.d.f., and likewise histograms of data sets that might be adequately modelled by the normal distribution, is 'bell-shaped'. Note that there is more than one normal distribution. No single distribution could possibly describe both the data of Figure 5.1, which have their mode around zero and which vary from about -4 minutes of arc to over 5 minutes, and those of Figure 5.2, whose mode is at about 40inches and which range from approximately 33inches to 48inches. In the real world there are many instances of random variation following this kind of pattern: the mode and the range of observed values will alter from random variable to random variable, but the characteristic bell shape of the data will be apparent. The four probability density functions shown in Figure 5.4 all correspond to different normal distributions. Figure 5.4 Four normal densities What has been described is another family of probability models, just like the binomial family (with two parameters, n and p) and the Poisson family (with one parameter, the mean p). The normal family has two parameters, one specifying location (the centre of the distribution) and one describing the degree of dispersion. In Chapter 2 the location parameter was denoted by p Elements of Statistics and the dispersion parameter was denoted by a; in fact, the parameter p is the mean of the normal distribution and a is its standard deviation. This information may be summarized as follows. The probability density function for the normal family of random variables is also given. The normal probability density function If the continuous random variable X is normally distributed with mean p and standard deviation a (variance a') then this may be written X N(P,0'); the probability density function of X is given by 1 1 2-11 f(z) = ~exp[-i (a)2] , -W < X < W. (5.1) As remarked already, you do not need to remember this formula in order to calculate normal A sketch of the p.d.f. of X is as follows. probabilities. The shape of the density function of X is often called 'bell-shaped'. The p.d.f. of X is positive for all values of X; however, observations more than about three standard deviations away from the mean are rather unlikely. The total area under the curve is 1. There are very few random variables for which possible observations include all negative and positive numbers. But for the normal distribution, extreme values may be regarded as occurring with negligible probability. One should not say 'the variation in Scottish chest measurements is normally distributed with mean 40 inches and standard deviation about 2 inches' (the implication being that negative observations are possible); rather, say 'the variation in Scottish chest measurements may be adequately modelled by a normal distri- bution with mean 40 inches and standard deviation 2 inches'. In the rest of this chapter we shall see many more applications in the real world where different members of the normal family provide good models of variation. But first, we shall explore some of the history of the development of the normal distribution. 186 Chapter 5 Section 5.1 Exercise 5.1 Without attempting geometrical calculations, suggest values for the par- ameters p and U for each of the normal probability densities that are shown in Figure 5.4. An early history Although the terminology was not standardized until after 1900, the normal distribution itself was certainly known before then (under a variety of dif- ferent names). The following is a brief account of the history of the normal distribution before the twentieth century. Credit for the very first appearance of the normal p.d.f. goes to Abraham de Moivre (1667-1754), a Protestant Frenchman who emigrated to London in 1688 to avoid religious persecution and lived there for the rest of his life, becoming an eminent mathematician. Prompted by a desire to compute the probabilities of winning in various games of chance, de Moivre obtained what is now recognized as the normal p.d.f., an approximation to a binomial prob- ability function (these were early days in the history of the binomial distri- bution). The pamphlet that contains this work was published in 1733.
Load more

Recommended publications
  • Effect of Probability Distribution of the Response Variable in Optimal Experimental Design with Applications in Medicine †
    mathematics Article Effect of Probability Distribution of the Response Variable in Optimal Experimental Design with Applications in Medicine † Sergio Pozuelo-Campos *,‡ , Víctor Casero-Alonso ‡ and Mariano Amo-Salas ‡ Department of Mathematics, University of Castilla-La Mancha, 13071 Ciudad Real, Spain; [email protected] (V.C.-A.); [email protected] (M.A.-S.) * Correspondence: [email protected] † This paper is an extended version of a published conference paper as a part of the proceedings of the 35th International Workshop on Statistical Modeling (IWSM), Bilbao, Spain, 19–24 July 2020. ‡ These authors contributed equally to this work. Abstract: In optimal experimental design theory it is usually assumed that the response variable follows a normal distribution with constant variance. However, some works assume other probability distributions based on additional information or practitioner’s prior experience. The main goal of this paper is to study the effect, in terms of efficiency, when misspecification in the probability distribution of the response variable occurs. The elemental information matrix, which includes information on the probability distribution of the response variable, provides a generalized Fisher information matrix. This study is performed from a practical perspective, comparing a normal distribution with the Poisson or gamma distribution. First, analytical results are obtained, including results for the linear quadratic model, and these are applied to some real illustrative examples. The nonlinear 4-parameter Hill model is next considered to study the influence of misspecification in a Citation: Pozuelo-Campos, S.; dose-response model. This analysis shows the behavior of the efficiency of the designs obtained in Casero-Alonso, V.; Amo-Salas, M.
    [Show full text]
  • A Matrix-Valued Bernoulli Distribution Gianfranco Lovison1 Dipartimento Di Scienze Statistiche E Matematiche “S
    Journal of Multivariate Analysis 97 (2006) 1573–1585 www.elsevier.com/locate/jmva A matrix-valued Bernoulli distribution Gianfranco Lovison1 Dipartimento di Scienze Statistiche e Matematiche “S. Vianelli”, Università di Palermo Viale delle Scienze 90128 Palermo, Italy Received 17 September 2004 Available online 10 August 2005 Abstract Matrix-valued distributions are used in continuous multivariate analysis to model sample data matrices of continuous measurements; their use seems to be neglected for binary, or more generally categorical, data. In this paper we propose a matrix-valued Bernoulli distribution, based on the log- linear representation introduced by Cox [The analysis of multivariate binary data, Appl. Statist. 21 (1972) 113–120] for the Multivariate Bernoulli distribution with correlated components. © 2005 Elsevier Inc. All rights reserved. AMS 1991 subject classification: 62E05; 62Hxx Keywords: Correlated multivariate binary responses; Multivariate Bernoulli distribution; Matrix-valued distributions 1. Introduction Matrix-valued distributions are used in continuous multivariate analysis (see, for exam- ple, [10]) to model sample data matrices of continuous measurements, allowing for both variable-dependence and unit-dependence. Their potentials seem to have been neglected for binary, and more generally categorical, data. This is somewhat surprising, since the natural, elementary representation of datasets with categorical variables is precisely in the form of sample binary data matrices, through the 0-1 coding of categories. E-mail address: [email protected] URL: http://dssm.unipa.it/lovison. 1 Work supported by MIUR 60% 2000, 2001 and MIUR P.R.I.N. 2002 grants. 0047-259X/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2005.06.008 1574 G.
    [Show full text]
  • Lecture.7 Poisson Distributions - Properties, Normal Distributions- Properties
    Lecture.7 Poisson Distributions - properties, Normal Distributions- properties Theoretical Distributions Theoretical distributions are 1. Binomial distribution Discrete distribution 2. Poisson distribution 3. Normal distribution Continuous distribution Discrete Probability distribution Bernoulli distribution A random variable x takes two values 0 and 1, with probabilities q and p ie., p(x=1) = p and p(x=0)=q, q-1-p is called a Bernoulli variate and is said to be Bernoulli distribution where p and q are probability of success and failure. It was given by Swiss mathematician James Bernoulli (1654-1705) Example • Tossing a coin(head or tail) • Germination of seed(germinate or not) Binomial distribution Binomial distribution was discovered by James Bernoulli (1654-1705). Let a random experiment be performed repeatedly and the occurrence of an event in a trial be called as success and its non-occurrence is failure. Consider a set of n independent trails (n being finite), in which the probability p of success in any trail is constant for each trial. Then q=1-p is the probability of failure in any trail. 1 The probability of x success and consequently n-x failures in n independent trails. But x successes in n trails can occur in ncx ways. Probability for each of these ways is pxqn-x. P(sss…ff…fsf…f)=p(s)p(s)….p(f)p(f)…. = p,p…q,q… = (p,p…p)(q,q…q) (x times) (n-x times) Hence the probability of x success in n trials is given by x n-x ncx p q Definition A random variable x is said to follow binomial distribution if it assumes non- negative values and its probability mass function is given by P(X=x) =p(x) = x n-x ncx p q , x=0,1,2…n q=1-p 0, otherwise The two independent constants n and p in the distribution are known as the parameters of the distribution.
    [Show full text]
  • 1 One Parameter Exponential Families
    1 One parameter exponential families The world of exponential families bridges the gap between the Gaussian family and general dis- tributions. Many properties of Gaussians carry through to exponential families in a fairly precise sense. • In the Gaussian world, there exact small sample distributional results (i.e. t, F , χ2). • In the exponential family world, there are approximate distributional results (i.e. deviance tests). • In the general setting, we can only appeal to asymptotics. A one-parameter exponential family, F is a one-parameter family of distributions of the form Pη(dx) = exp (η · t(x) − Λ(η)) P0(dx) for some probability measure P0. The parameter η is called the natural or canonical parameter and the function Λ is called the cumulant generating function, and is simply the normalization needed to make dPη fη(x) = (x) = exp (η · t(x) − Λ(η)) dP0 a proper probability density. The random variable t(X) is the sufficient statistic of the exponential family. Note that P0 does not have to be a distribution on R, but these are of course the simplest examples. 1.0.1 A first example: Gaussian with linear sufficient statistic Consider the standard normal distribution Z e−z2=2 P0(A) = p dz A 2π and let t(x) = x. Then, the exponential family is eη·x−x2=2 Pη(dx) / p 2π and we see that Λ(η) = η2=2: eta= np.linspace(-2,2,101) CGF= eta**2/2. plt.plot(eta, CGF) A= plt.gca() A.set_xlabel(r'$\eta$', size=20) A.set_ylabel(r'$\Lambda(\eta)$', size=20) f= plt.gcf() 1 Thus, the exponential family in this setting is the collection F = fN(η; 1) : η 2 Rg : d 1.0.2 Normal with quadratic sufficient statistic on R d As a second example, take P0 = N(0;Id×d), i.e.
    [Show full text]
  • Discrete Probability Distributions Uniform Distribution Bernoulli
    Discrete Probability Distributions Uniform Distribution Experiment obeys: all outcomes equally probable Random variable: outcome Probability distribution: if k is the number of possible outcomes, 1 if x is a possible outcome p(x)= k ( 0 otherwise Example: tossing a fair die (k = 6) Bernoulli Distribution Experiment obeys: (1) a single trial with two possible outcomes (success and failure) (2) P trial is successful = p Random variable: number of successful trials (zero or one) Probability distribution: p(x)= px(1 − p)n−x Mean and variance: µ = p, σ2 = p(1 − p) Example: tossing a fair coin once Binomial Distribution Experiment obeys: (1) n repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: number of successful trials n x n−x Probability distribution: b(x; n,p)= x p (1 − p) Mean and variance: µ = np, σ2 = np(1 − p) Example: tossing a fair coin n times Approximations: (1) b(x; n,p) ≈ p(x; λ = pn) if p ≪ 1, x ≪ n (Poisson approximation) (2) b(x; n,p) ≈ n(x; µ = pn,σ = np(1 − p) ) if np ≫ 1, n(1 − p) ≫ 1 (Normal approximation) p Geometric Distribution Experiment obeys: (1) indeterminate number of repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: trial number of first successful trial Probability distribution: p(x)= p(1 − p)x−1 1 2 1−p Mean and variance: µ = p , σ = p2 Example: repeated attempts to start
    [Show full text]
  • STATS8: Introduction to Biostatistics 24Pt Random Variables And
    STATS8: Introduction to Biostatistics Random Variables and Probability Distributions Babak Shahbaba Department of Statistics, UCI Random variables • In this lecture, we will discuss random variables and their probability distributions. • Formally, a random variable X assigns a numerical value to each possible outcome (and event) of a random phenomenon. • For instance, we can define X based on possible genotypes of a bi-allelic gene A as follows: 8 < 0 for genotype AA; X = 1 for genotype Aa; : 2 for genotype aa: • Alternatively, we can define a random, Y , variable this way: 0 for genotypes AA and aa; Y = 1 for genotype Aa: Random variables • After we define a random variable, we can find the probabilities for its possible values based on the probabilities for its underlying random phenomenon. • This way, instead of talking about the probabilities for different outcomes and events, we can talk about the probability of different values for a random variable. • For example, suppose P(AA) = 0:49, P(Aa) = 0:42, and P(aa) = 0:09. • Then, we can say that P(X = 0) = 0:49, i.e., X is equal to 0 with probability of 0.49. • Note that the total probability for the random variable is still 1. Random variables • The probability distribution of a random variable specifies its possible values (i.e., its range) and their corresponding probabilities. • For the random variable X defined based on genotypes, the probability distribution can be simply specified as follows: 8 < 0:49 for x = 0; P(X = x) = 0:42 for x = 1; : 0:09 for x = 2: Here, x denotes a specific value (i.e., 0, 1, or 2) of the random variable.
    [Show full text]
  • ONE SAMPLE TESTS the Following Data Represent the Change
    1 WORKED EXAMPLES 6 INTRODUCTION TO STATISTICAL METHODS EXAMPLE 1: ONE SAMPLE TESTS The following data represent the change (in ml) in the amount of Carbon monoxide transfer (an indicator of improved lung function) in smokers with chickenpox over a one week period: 33, 2, 24, 17, 4, 1, -6 Is there evidence of significant improvement in lung function (a) if the data are normally distributed with σ = 10, (b) if the data are normally distributed with σ unknown? Use a significance level of α = 0.05. SOLUTION: (a) Here we have a sample of size 7 with sample mean x = 10.71. We want to test H0 : μ = 0.0, H1 : μ = 0.0, 6 under the assumption that the data follow a Normal distribution with σ = 10.0 known. Then, we have, in the Z-test, 10.71 0.0 z = − = 2.83, 10.0/√7 which lies in the critical region, as the critical values for this test are 1.96, for significance ± level α = 0.05. Therefore we have evidence to reject H0. The p-value is given by p = 2Φ( 2.83) = 0.004 < α. − (b) The sample variance is s2 = 14.192. In the T-test, we have test statistic t given by x 0.0 10.71 0.0 t = − = − = 2.00. s/√n 14.19/√7 The upper critical value CR is obtained by solving FSt(n 1)(CR) = 0.975, − where FSt(n 1) is the cdf of a Student-t distribution with n 1 degrees of freedom; here n = 7, so − − we can use statistical tables or a computer to find that CR = 2.447, and note that, as Student-t distributions are symmetric the lower critical value is CR.
    [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]
  • Poisson Versus Negative Binomial Regression
    Handling Count Data The Negative Binomial Distribution Other Applications and Analysis in R References Poisson versus Negative Binomial Regression Randall Reese Utah State University [email protected] February 29, 2016 Randall Reese Poisson and Neg. Binom Handling Count Data The Negative Binomial Distribution Other Applications and Analysis in R References Overview 1 Handling Count Data ADEM Overdispersion 2 The Negative Binomial Distribution Foundations of Negative Binomial Distribution Basic Properties of the Negative Binomial Distribution Fitting the Negative Binomial Model 3 Other Applications and Analysis in R 4 References Randall Reese Poisson and Neg. Binom Handling Count Data The Negative Binomial Distribution ADEM Other Applications and Analysis in R Overdispersion References Count Data Randall Reese Poisson and Neg. Binom Handling Count Data The Negative Binomial Distribution ADEM Other Applications and Analysis in R Overdispersion References Count Data Data whose values come from Z≥0, the non-negative integers. Classic example is deaths in the Prussian army per year by horse kick (Bortkiewicz) Example 2 of Notes 5. (Number of successful \attempts"). Randall Reese Poisson and Neg. Binom Handling Count Data The Negative Binomial Distribution ADEM Other Applications and Analysis in R Overdispersion References Poisson Distribution Support is the non-negative integers. (Count data). Described by a single parameter λ > 0. When Y ∼ Poisson(λ), then E(Y ) = Var(Y ) = λ Randall Reese Poisson and Neg. Binom Handling Count Data The Negative Binomial Distribution ADEM Other Applications and Analysis in R Overdispersion References Acute Disseminated Encephalomyelitis Acute Disseminated Encephalomyelitis (ADEM) is a neurological, immune disorder in which widespread inflammation of the brain and spinal cord damages tissue known as white matter.
    [Show full text]
  • 6.1 Definition: the Density Function of Th
    CHAPTER 6: Some Continuous Probability Distributions Continuous Uniform Distribution: 6.1 Definition: The density function of the continuous random variable X on the interval [A; B] is 1 A x B B A ≤ ≤ f(x; A; B) = 8 − < 0 otherwise: : Application: Some continuous random variables in the physical, management, and biological sciences have approximately uniform probability distributions. For example, suppose we are counting events that have a Poisson distribution, such as telephone calls coming into a switchboard. If it is known that exactly one such event has occurred in a given interval, say (0; t),then the actual time of occurrence is distributed uniformly over this interval. Example: Arrivals of customers at a certain checkout counter follow a Poisson distribution. It is known that, during a given 30-minute period, one customer arrived at the counter. Find the probability that the customer arrived during the last 5 minutes of the 30-minute period. Solution: As just mentioned, the actual time of arrival follows a uniform distribution over the interval of (0; 30). If X denotes the arrival time, then 30 1 30 25 1 P (25 X 30) = dx = − = ≤ ≤ Z25 30 30 6 Theorem 6.1: The mean and variance of the uniform distribution are 2 B 2 2 B 1 x B A A+B µ = A x B A = 2(B A) = 2(B− A) = 2 : R − h − iA − It is easy to show that (B A)2 σ2 = − 12 Normal Distribution: 6.2 Definition: The density function of the normal random variable X, with mean µ and variance σ2, is 2 1 (1=2)[(x µ)/σ] n(x; µ, σ) = e− − < x < ; p2πσ − 1 1 where π = 3:14159 : : : and e = 2:71828 : : : Example: The SAT aptitude examinations in English and Mathematics were originally designed so that scores would be approximately normal with µ = 500 and σ = 100.
    [Show full text]
  • Lecture 6: Special Probability Distributions
    The Discrete Uniform Distribution The Bernoulli Distribution The Binomial Distribution The Negative Binomial and Geometric Di Lecture 6: Special Probability Distributions Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics Istanbul˙ K¨ult¨ur University The Discrete Uniform Distribution The Bernoulli Distribution The Binomial Distribution The Negative Binomial and Geometric Di Outline 1 The Discrete Uniform Distribution 2 The Bernoulli Distribution 3 The Binomial Distribution 4 The Negative Binomial and Geometric Distribution The Discrete Uniform Distribution The Bernoulli Distribution The Binomial Distribution The Negative Binomial and Geometric Di Outline 1 The Discrete Uniform Distribution 2 The Bernoulli Distribution 3 The Binomial Distribution 4 The Negative Binomial and Geometric Distribution The Discrete Uniform Distribution The Bernoulli Distribution The Binomial Distribution The Negative Binomial and Geometric Di The Discrete Uniform Distribution If a random variable can take on k different values with equal probability, we say that it has a discrete uniform distribution; symbolically, Definition 1 A random variable X has a discrete uniform distribution and it is referred to as a discrete uniform random variable if and only if its probability distribution is given by 1 f (x)= for x = x , x , ··· , xk k 1 2 where xi = xj when i = j. The Discrete Uniform Distribution The Bernoulli Distribution The Binomial Distribution The Negative Binomial and Geometric Di The mean and the variance of this distribution are Mean: k k 1 x1 + x2 + ···+ xk μ = E[X ]= xi f (xi ) = xi · = , k k i=1 i=1 1/k Variance: k 2 2 2 1 σ = E[(X − μ) ]= (xi − μ) · k i=1 2 2 2 (x − μ) +(x − μ) + ···+(xk − μ) = 1 2 k The Discrete Uniform Distribution The Bernoulli Distribution The Binomial Distribution The Negative Binomial and Geometric Di In the special case where xi = i, the discrete uniform distribution 1 becomes f (x)= k for x =1, 2, ··· , k, and in this from it applies, for example, to the number of points we roll with a balanced die.
    [Show full text]
  • Statistical Inference
    GU4204: Statistical Inference Bodhisattva Sen Columbia University February 27, 2020 Contents 1 Introduction5 1.1 Statistical Inference: Motivation.....................5 1.2 Recap: Some results from probability..................5 1.3 Back to Example 1.1...........................8 1.4 Delta method...............................8 1.5 Back to Example 1.1........................... 10 2 Statistical Inference: Estimation 11 2.1 Statistical model............................. 11 2.2 Method of Moments estimators..................... 13 3 Method of Maximum Likelihood 16 3.1 Properties of MLEs............................ 20 3.1.1 Invariance............................. 20 3.1.2 Consistency............................ 21 3.2 Computational methods for approximating MLEs........... 21 3.2.1 Newton's Method......................... 21 3.2.2 The EM Algorithm........................ 22 1 4 Principles of estimation 23 4.1 Mean squared error............................ 24 4.2 Comparing estimators.......................... 25 4.3 Unbiased estimators........................... 26 4.4 Sufficient Statistics............................ 28 5 Bayesian paradigm 33 5.1 Prior distribution............................. 33 5.2 Posterior distribution........................... 34 5.3 Bayes Estimators............................. 36 5.4 Sampling from a normal distribution.................. 37 6 The sampling distribution of a statistic 39 6.1 The gamma and the χ2 distributions.................. 39 6.1.1 The gamma distribution..................... 39 6.1.2 The Chi-squared distribution.................. 41 6.2 Sampling from a normal population................... 42 6.3 The t-distribution............................. 45 7 Confidence intervals 46 8 The (Cramer-Rao) Information Inequality 51 9 Large Sample Properties of the MLE 57 10 Hypothesis Testing 61 10.1 Principles of Hypothesis Testing..................... 61 10.2 Critical regions and test statistics.................... 62 10.3 Power function and types of error.................... 64 10.4 Significance level............................
    [Show full text]