Continuous Random Variables and Probability Distributions

In Chap. 2, we discussed the basic concepts of probability, discrete and continuous random variables, univariate and joint probability distributions, conditional probability, and independence. A reasonably complete and self-sustained treatment has been given to these topics, which form a theoretical basis for practical statistical data analysis. This chapter discusses statistical analysis based on available sample data. Specifically, it introduces statistical procedures to determine an appropriate probability distribution for a random variable based on a limited set of sample data. Reliability analysis that will be discussed in subsequent chapters often requires these procedures. In this chapter, discussion of statistical data analysis will be separately conducted on two different branches of techniques: (i) conventional statistical methods (graphical methods and statistical hypothesis tests) and (ii) Bayesian statistics.

Whether an experiment yields qualitative or quantitative outcomes, methods of statistical analysis require that we focus on certain numerical aspects of the data (such as a sample proportion x/n, mean \( \bar{x} \), or standard deviation s). The concept of a random variable allows us to pass from the experimental outcomes themselves to a numerical function of the outcomes. There are two fundamentally different types of random variables—discrete random variables and continuous random variables. In this chapter, we examine the basic properties and discuss the most important examples of discrete variables. Chapter 4 focuses on continuous random variables.KeywordsNegative BinomialSingle SpikeletBinomial ExperimentUnacceptable VoltageAcceptance Sampling PlansThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The problem of logarithmically asymptotically optimal hypothesis testing for a continuous random variable (CRV) is considered.It is assumed that M continuous probability distributions (CPDs) are known, and the object described by the CRV follows one of them.This problem, in the case of discrete probability distributions, was introduced and extensively studied in [1].In this paper, a quantization method is applied to continuous distributions.Some known results are generalized and reformulated for the continuous case.

In many real-world applications, the phenomena of interest are continuous in nature and modeled through continuous probability distributions, but their observed values are actually discrete and hence it would be more reasonable and convenient to choose an appropriate (multivariate) discrete distribution generated from the underlying continuous model preserving one or more important features. In this paper, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely (1) the joint density function, or (2) the joint survival function. Examples of applications are presented, which involve two types of bivariate exponential distributions, in order to illustrate how the discretization procedures work and show whether and to which extent they alter the dependence structure of the original model. We also prove that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. Two real datasets, considering correlated discrete recurrence times (the former) and counts (the latter) are eventually fitted using two discrete analogues of a bivariate exponential distribution.

The survival function method is a well-known alternative procedure for calculating the expectation of a nonnegative random variable with a continuous probability distribution. This paper proves a generalization of this method that can be used to calculate the expectation of a continuously differentiable function of a nonnegative random variable with a continuous, discrete or mixed probability distribution. A similar result is used in actuarial probability manuals, but proofs for the general case along with required assumptions are often not provided. Moreover, generalizations of the survival function method are typically not mentioned in probability textbooks. One of the main contributions of this paper is that it explores some technical conditions that guarantee that the method can be applied. This paper also provides an alternate proof of the generalized survival function method along with a different set of conditions under which the method can be proved to hold. Finally, examples of how the method can be applied to calculate expectations and moment-generating functions and to derive an integral identity are given.

In many management science or economic applications, it is common to represent the key uncertain inputs as continuous random variables. However, when analytic techniques fail to provide a closed-form solution to a problem or when one needs to reduce the computational load, it is often necessary to resort to some problem-specific approximation technique or approximate each given continuous probability distribution by a discrete distribution. Many discretization methods have been proposed so far; in this work, we revise the most popular techniques, highlighting their strengths and weaknesses, and empirically investigate their performance through a comparative study applied to a well-known engineering problem, formulated as a stress–strength model, with the aim of weighting up their feasibility and accuracy in recovering the value of the reliability parameter, also with reference to the number of discrete points. The results overall reward a recently introduced method as the best performer, which derives the discrete approximation as the numerical solution of a constrained non-linear optimization, preserving the first two moments of the original distribution. This method provides more accurate results than an ad-hoc first-order approximation technique. However, it is the most computationally demanding as well and the computation time can get even larger than that required by Monte Carlo approximation if the number of discrete points exceeds a certain threshold.

In this paper we study the covariance structure of the number of nodes k and l steps away from the root in random recursive trees. We give an analytic expression valid for all k, l and tree sizes N. The fraction of nodes k steps away from the root is a random probability distribution in k. The expression for the covariances allows us to show that the total variation distance between this (random) probability distribution and its mean converges in probability to zero. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 519–539, 2002

What is Statistics? Describing Sets of Measurements Probability Random Variables and Probability Distributions Three Useful Discrete Probability Distributions The Normal and Other Continuous Probability Distributions Sampling and Sampling Distributions Large sample Estimation Large Sample Tests of Hypotheses Inferences from Small Samples The Analysis of Variance Linear Regression and Correlation Multiple Linear Regression Elements of Time Series Analysis Forecasting Models Quality Control Survey Sampling Analysis of Enumerative Data Nonparametric Statistics Decision Analysis.

This chapter reviews some subjects in mathematics and physics that are used in different contexts throughout this book. The selection of subjects and the level of their coverage reflect the author’s perception of what potential users of this text were exposed to in their earlier studies. Therefore, only brief overview is given of some subjects while somewhat more comprehensive discussion is given of others. In neither case can the coverage provided substitute for the actual learning of these subjects that are covered in detail by many textbooks. A random variable is an observable whose repeated determination yields a series of numerical values (“realizations” of the random variable) that vary from trial to trial in a way characteristic of the observable. The outcomes of tossing a coin or throwing a die are familiar examples of discrete random variables. The position of a dust particle in air and the lifetime of a light bulb are continuous random variables. Discrete random variables are characterized by probability distributions; Pn denotes the probability that a realization of the given random variable is n. Continuous random variables are associated with probability density functions P(x): P(x1)dx denotes the probability that the realization of the variable x will be in the interval x1 . . . x1+dx.

After a brief expository account of the Shannon-Jaynes principle of maximum entropy (POME) for discrete and continuous variables, we give here an account of some recent research work which (i) a “histogram” method to contrast the discrete and continuous modes computation and the role of histogram in actual practice when dealing with continuous probability distributions. (ii) The idea of mean logarithmic decrement associated with a probability distribution is introduced and is shown to be related to the concept of differential entropy. The mean with respect to an arbitrary probability distribution of the logarithm of the ratio of the probability density function for an exponential distribution is discussed in the context of hydrological investigations. Unlike the entropy of a continuous probability distribution introduced in (i), this quantity which is an example of the Kullback-Leibler (KL) Information, is always positive and invariant under coordinate transformation. (iii) The constraints entering into POME as well as a minimum K L information are identified as a class of sufficient statistics which determine the unknown parameters in the probability density functions that occur in the most commonly used hydrological models. (iv) An example of (iii) where only the first two moments in a semi-infinite domain are givens is discussed to shed light on the limitations of POME, recentky recognized by Wragg and coworkers, and is made relevant to the work of Sonuga on rainfall-run off relationship. Finally, (v) a method of generating probability distributions starting from one basic distribution employing coordinate transformations is given. This in conjuction with (iii) leads to the notions of “Physical constraints” in contrast to the “mathematical constraints” in examing parameter estimation.

Continuous probability distribution was one of the mathematics concept that students had difficulty.This study analyzed the definition and introduction of the continuous probability distribution under the 2009-revised curriculum and 2015-revised curriculum.In this study, the following subjects were studied.Firstly, definitions of continuous probability variable in ' Probability and Statistics' textbook developed under the 2009-revised curriculum and 2015-revised curriculum were analyzed.Secondly, introductions of continuous probability distribution in ' Probability and Statistics' textbook developed under the 2009-revised curriculum and 2015-revised curriculum were analyzed.The results of this study were as follows.First, 8 textbooks under the 2009-revised curriculum defined the continuous probability variable as probability variable with all the real values within a range or an interval.And 1 textbook under the 2009-revised curriculum defined the continuous probability variable as probability variable when the set of its value is uncountable.But all textbooks under the 2015-revised curriculum defined the continuous probability variable as probability variable with all the real values within a range.Second, 4 textbooks under the 2009-revised curriculum and 4 textbooks under 2015-revised curriculum introduced a continuous random distribution using an uniformly distribution.And 5 textbooks under the 2009-revised curriculum and 5 textbooks under the 2015-revised curriculum introduced a continuous random distribution using a relative frequency density.

We propose an original classification of several discrete and continuous probability distributions. We establish links between these distributions, in particular the little known relationship between the negative hypergeometric distribution and the beta distribution. These relations allow us to propose a structure of relations which is summarized in graphic form.Our classification emphasises the analogy between certain discrete and continuous distributions. This analogy makes it possible to establish relations between the theory of point processes and the theory of survey sampling. It also makes it possible to envisage the use of link functions that are little used in generalised regression.

The ultimate goal of our study is to obtain suggestions for the development of devices capable of the automatic modeling of natural phenomena. It is therefore of fundamental importance to develop a theoretical basis for the description of their optimal performance. In the previous chapters it was stated that an empirical modeling of natural phenomena includes three main tasks: Estimation, storage, and application of a probability distribution. Each of these tasks can be optimized by using the methods described in the previous chapter. The aim of this section is to present the problems and the solutions related with an optimal storage of empirical information about a continuous probability distribution in a system comprised of a finite number of discrete memory units. Such a system can be considered as a basic building block of an automatic modeler of natural phenomena. For example, we can imagine the brain of a biological organism or the digital memory of a computer that continually obtains signals from its surroundings and it optimally stores the corresponding empirical information. The first problem is the estimation of the probability density function of a continuous variable from the empirical data. We have already seen that this can be solved using Parzen’s window. [2] The second problem is the storage of the continuous probability density in a discrete system. This task is generally related with a loss of information and thus the question arises, how one can minimize this loss.KeywordsWindow FunctionExcited NeuronReference VectorDiscrete Random VariableMemory UnitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper introduces a new class of continuous probability distributions that are flexible enough to represent a wide range of uncertainties such as those that commonly arise in business, technology, and science. In many such cases, the nature of the uncertainty is more naturally characterized by quantiles than by parameters of familiar continuous probability distributions. In the practice of decision analysis, it is common to fit a hand-drawn curve to quantile outputs from probability elicitations on a continuous uncertain quantity and to then discretize the curve. The resulting discrete probability distribution is an approximation that cuts off the distribution's tails and eliminates intermediate values. Quantile-parameterized distributions address this problem by using quantiles themselves to parameterize a continuous probability distribution. We define quantile-parameterized distributions, illustrate their flexibility and range of applicability, and conclude with practical considerations when parameterizing distributions using inconsistent quantile assessments.

At first in Chap. 3 are described such important notions as characteristic functions of a probability distribution defined as Fourier transform of a probability density function and constants characterizing random variables: an average value, a median value describing the middle of a distribution, a variation ratio closely connected with a standard deviation. Other characteristic constants characterizing random variables are: an asymmetry coefficient and an excess coefficient. Other parameters describing shape of a probability density functions called quantiles are also defined. A number of useful probability distributions are presented, among them are discrete probability distributions: a binomial distribution, a multinomial distribution, a Poisson distribution, and continuous probability distributions: a rectangular distribution, an exponential distribution, a Weibull distribution and a Student distribution are described. Finally are given some introductory remarks concerning multidimensional probability distributions, developed further in Chaps. 5 and 7 for analyzing various practical problems.