Random Variables and Probability

2021 ◽  
pp. 109-124
Author(s):  
Timothy E. Essington
Keyword(s):  
Parameter Estimation ◽  
Model Selection ◽  
Beta Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Density Functions ◽  
The Core ◽  
Discrete Random Variables ◽  
Probability Mass ◽  
T Distribution

The chapter “Random Variables and Probability” serves as both a review and a reference on probability. The random variable is the core concept in understanding probability, parameter estimation, and model selection. This chapter reviews the basic idea of a random variable and discusses the two main kinds of random variables: discrete random variables and continuous random variables. It covers the distinction between discrete and continuous random variables and outlines the most common probability mass or density functions used in ecology. Advanced sections cover distributions such as the gamma distribution, Student’s t-distribution, the beta distribution, the beta-binomial distribution, and zero-inflated models.

Probability and random variables

2019 ◽  
pp. 38-59
Author(s):  
M. D. Edge
Keyword(s):  
Probability Density ◽  
Random Variables ◽  
Bayes Theorem ◽  
Probability Density Functions ◽  
Density Functions ◽  
Conditional Probabilities ◽  
Discrete Random Variables ◽  
New Information ◽  
Probability Mass ◽  
Mass Functions

This chapter considers the rules of probability. Probabilities are non-negative, they sum to one, and the probability that either of two mutually exclusive events occurs is the sum of the probability of the two events. Two events are said to be independent if the probability that they both occur is the product of the probabilities that each event occurs. Bayes’ theorem is used to update probabilities on the basis of new information, and it is shown that the conditional probabilities P(A|B) and P(B|A) are not the same. Finally, the chapter discusses ways in which distributions of random variables can be described, using probability mass functions for discrete random variables and probability density functions for continuous random variables.

Probability Mass Functions

2019 ◽  
pp. 87-107
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Bayesian Inference ◽  
Binomial Distribution ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Posterior Distributions ◽  
Bernoulli Distribution ◽  
Discrete Random Variables ◽  
Probability Mass ◽  
Mass Functions

This chapter focuses on probability mass functions. One of the primary uses of Bayesian inference is to estimate parameters. To do so, it is necessary to first build a good understanding of probability distributions. This chapter introduces the idea of a random variable and presents general concepts associated with probability distributions for discrete random variables. It starts off by discussing the concept of a function and goes on to describe how a random variable is a type of function. The binomial distribution and the Bernoulli distribution are then used as examples of the probability mass functions (pmf’s). The pmfs can be used to specify prior distributions, likelihoods, likelihood profiles and/or posterior distributions in Bayesian inference.

Review Of Some Mathematical And Physical Subjects

2006 ◽  
Author(s):  
Abraham Nitzan
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Probability Density Functions ◽  
Density Functions ◽  
Discrete Random Variables ◽  
Actual Learning ◽  
The Given ◽  
Selection Of ◽  
Comprehensive Discussion

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.

Parameter Estimation

2011 ◽  
pp. 172-180
Author(s):  
Karl-Ernst Biebler
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Random Variables ◽  
Random Variable ◽  
Likelihood Method ◽  
Approximate Methods ◽  
Normal Distributions ◽  
Parameter Estimations ◽  
Binomial Distributions ◽  
Starting Point

Parameters are numbers which characterize random variables. They make possible the summarizing description of the observations, serve as the basis of statistical decisions and are calculated from the data. Point estimations and confidence estimations are introduced. Samples of the observed random variable are a starting point. The maximum-likelihood method for the construction of parameter estimations is introduced here. Examples concern the normal distributions and the binomial distributions. Approximate methods of the parameter estimation also can be too inaccurate at large sample sizes. This is demonstrated in an example from genetics.

On the stationary distribution of some extremal Markovian sequences

1990 ◽  
Vol 27 (2) ◽  
pp. 291-302 ◽  
Author(s):  
M. T. Alpuim ◽  
E. Athayde
Keyword(s):  
Stationary Distribution ◽  
Beta Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Stationary Distributions

This paper is concerned with the Markovian sequence Xn = Zn max{Xn–1, Yn},n ≧ 1, where X0 is any random variable, {Zn} and {Yn} are independent sequences of i.i.d. random variables both independent of X0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn's are random variables concentrated on the interval [0, 1], namely having a beta distribution.

A storage process by semi-Markov input

1975 ◽  
Vol 7 (4) ◽  
pp. 830-844 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Recurrence Formula ◽  
Random Variables ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Storage Process ◽  
Markov Sequence ◽  
Discrete Random Variables ◽  
Matrix Factorisation

A sequence of random variables η0, η1, …, ηn, … is defined by the recurrence formula ηn = max (ηn–1 + ξn, 0) where η0 is a discrete random variable taking on non-negative integers only and ξ1, ξ2, … ξn, … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.

scholarly journals On Some Compound Random Variables Motivated by Bulk Queues

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Romeo Meštrović
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Time Of Arrival ◽  
Uniform Random Variable ◽  
Wide Range ◽  
Bulk Queue ◽  
Probability Mass ◽  
Number Of Customers

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group (Y(t)) and its size (X) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variableQ(t)which expresses the number of customers that arrive in this bulk queue during any considered periodt. Notice thatQ(t)can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributionsQ(t)concerning certain pairs(Y(t),X)of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases whenY(t)and/orXare some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.

On the stationary distribution of some extremal Markovian sequences

1990 ◽  
Vol 27 (02) ◽  
pp. 291-302 ◽  
Author(s):  
M. T. Alpuim ◽  
E. Athayde
Keyword(s):  
Stationary Distribution ◽  
Beta Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Stationary Distributions

This paper is concerned with the Markovian sequence Xn = Zn max{Xn– 1, Yn },n ≧ 1, where X 0 is any random variable, {Zn } and {Yn } are independent sequences of i.i.d. random variables both independent of X 0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn 's are random variables concentrated on the interval [0, 1], namely having a beta distribution.

Peluang, Peubah Acak Diskrit, dan Sebaran Peluang Peubah Acak Diskrit

2020 ◽  
Author(s):  
Ahmad Sudi Pratikno
Keyword(s):  
Probability Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Discrete Random Variables

Probability to learn someone's chance in getting or winning an event. In the discrete random variable is more identical to repeated experiments, to form a pattern. Discrete random variables can be calculated as the probability distribution by calculating each value that might get a certain probability value.

A storage process by semi-Markov input

1975 ◽  
Vol 7 (04) ◽  
pp. 830-844
Author(s):  
Lajos Takács
Keyword(s):  
Recurrence Formula ◽  
Random Variables ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Storage Process ◽  
Markov Sequence ◽  
Discrete Random Variables ◽  
Matrix Factorisation

A sequence of random variables η 0, η 1, …, ηn , … is defined by the recurrence formula ηn = max (η n–1 + ξn , 0) where η 0 is a discrete random variable taking on non-negative integers only and ξ 1, ξ 2, … ξn , … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.

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