Probability distributions and orthogonal parameters

1950 ◽  
Vol 46 (2) ◽  
pp. 281-284 ◽  
Author(s):  
V. S. Huzurbazar
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Orthogonal Parameters ◽  
Image Position

Let f(x, αi) be the probability density function of a distribution depending on n parameters αi(i = 1,2, …, n). Then following Jeffreys(1) we shall say that the parameters αi are orthogonal if

scholarly journals Maximum-likelihood estimation of the parameters of a four-parameter class of probability distributions

1988 ◽  
Vol 31 (2) ◽  
pp. 271-283 ◽  
Author(s):  
Siegfried H. Lehnigk
Keyword(s):  
Maximum Likelihood ◽  
Probability Density Function ◽  
Maximum Likelihood Estimation ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Likelihood Estimation ◽  
Initial Shape ◽  
Definition Of ◽  
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We shall concern ourselves with the class of continuous, four-parameter, one-sided probability distributions which can be characterized by the probability density function (pdf) classIt depends on the four parameters: shift c ∈ R, scale b > 0, initial shape p < 1, and terminal shape β > 0. For p ≦ 0, the definition of f(x) can be completed by setting f(c) = β/bΓ(β−1)>0 if p = 0, and f(c) = 0 if p < 0. For 0 < p < 1, f(x) remains undefined at x = c; f(x)↑ + ∞ as x↓c.

scholarly journals A Probability Density Function Generator Based on Deep Learning

2018 ◽  
Author(s):  
Chi-Hua Chen ◽  
Fangying Song ◽  
Feng-Jang Hwang ◽  
Ling Wu
Keyword(s):  
Deep Learning ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Real Data ◽  
Learning Model ◽  
Cumulative Distribution ◽  
Function Generator ◽  
Deep Learning Model

To generate a probability density function (PDF) for fitting probability distributions of real data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be adopted as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of trained deep learning model can be used to estimate the PDF. To evaluate the proposed method, numerical experiments with single and mixed distributions are performed. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method.

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
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In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

Deterministic epidemic waves

1976 ◽  
Vol 80 (2) ◽  
pp. 315-330 ◽  
Author(s):  
C. Atkinson ◽  
G. E. H. Reuter
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Deterministic Model ◽  
Removal Rate ◽  
Uniform Density ◽  
Space And Time ◽  
Time Variables ◽  
Image Position

In the well-known deterministic model for the spread of an epidemic, one considers a population of uniform density along a line and divides the population into three classes: susceptible but uninfected, infected and infectious, infected but removed. If we denote space and time variables by s, t and let x(s, t), y(s, t), z(s, t) be the proportions of the population at (s, t) in these three classes, then x + y + z = 1 and we suppose thatHere Ῡ(s, t) denotes a space average ∫ y(s + σ) p(σ) dσ, where p is a probability density function; b is the removal rate; the scale of t has been adjusted to remove a constant that would otherwise occur in (1).

scholarly journals FUNÇÃO DENSIDADE DE PROBABILIDADE APLICÁVEL À CIÊNCIA FLORESTAL

FLORESTA ◽  
2003 ◽  
Vol 33 (3) ◽  
Author(s):  
Eduardo Quadros Da Silva ◽  
Sylvio Péllico Netto ◽  
Sebastião Do Amaral Machado ◽  
Carlos Roberto Sanquetta
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Minas Gerais ◽  
Real Interval ◽  
Mathematical Function ◽  
Natural Forests ◽  
Normal Distribution Curve ◽  
Cariniana Legalis

Este trabalho tem como objetivo principal apresentar novas opções para o ajuste de distribuições de probabilidades que são utilizadas na Ciência Florestal. Alguns modelos contínuos apresentam certas distorções ao serem implementados no cálculo com dados oriundos de florestas naturais devido à grande variabilidade que se encontra nessas situações. Esse fato foi constatado principalmente quando se estudou a variável altura. Embora o modelo tenha sido construído com dados de alturas de florestas naturais, as fórmulas desenvolvidas poderão ser aplicadas para outras grandezas, principalmente se o gráfico apresentar assimetria, o que afasta a possibilidade de estudo por meio da distribuição normal. Neste estudo procurou-se mostrar que o modelo é adaptável também a dados de diâmetro e situações onde há simetria. Para a realização do trabalho inicialmente foram estudadas maneiras de modelar uma função matemática que pudesse ser transformada em função densidade de probabilidade. A função deveria assumir somente valores positivos, ser contínua e sua integral, considerando todo o intervalo real, deveria convergir para um. Foram feitas várias tentativas com funções matemáticas que, apesar de atenderem às condições de uma função densidade de probabilidade, não eram suficientemente flexíveis para se adaptar às características dos dados de uma floresta natural. Finalmente chegou-se a uma função que é definida por três sentenças, formada por um polinômio de grau n, uma curva crescente e uma curva decrescente positiva tendendo a zero com integral convergente no infinito. O polinômio explicou a maior parte dos dados e, para as classes onde este não produziu bom ajuste, foram elaboradas outras duas funções. Para os testes iniciais foram utilizados dados de alturas de Jequitibá-Rosa (Cariniana legalis), provenientes da Fazenda Reata, situada no município de Cássia, Minas Gerais. Para testar a aplicabilidade em outras situações procurou-se ajustar o modelo a dados de diâmetros e, após a aplicação do teste de Kolmogorov-Smirnov, os resultados mostraram-se satisfatórios. PROBABILITY DENSITY FUNCTION APPLICABLE TO FORESTRY Abstract The main objective of this research was to introduce new options for the fitting of probability distributions used in Forestry. Some continuous probability distributions present certain distortion when used in calculus with data from natural forests due to the high variability in such situations. This fact was especially noticed in studies of the variable height. Although the object of the study was natural forest’s height data, the developed formulas may be applied for other variables as well, especially if the resulted distribution is asymmetric which prevents the study to be made by the normal distribution curve. Before the study could be carried out, we did some work to model a mathematical function that could be changed into probability density function, that is, the functional values had to be positive, the function should be continuous, and its integral – considering the whole real interval – had to converge to one. Several attempts were made with mathematical functions that fulfilled the requirements of probability density function, but none was flexible enough to suit the data of a natural forest. Finally, a function was obtained which was defined by several sentences, formed by an n-polynomial, preferably a 5 degree increasing curve, a positive decreasing curve tending to zero, and the integral converging to the infinite. The polynomial explains most of the data; for the cases in which it fails to produce a good fitting, two other functions were created. The species used at first was Jequitibá-Rosa (Cariniana legalis), whose data came from a farm located in the municipality of Cássia, in Minas Gerais, Brazil. The total frequency observed for Jequitibá-Rosa, which was 493, was explained by the model with 492.9. The mean height achieved in the developed model 17.8 m is very close to the one that was calculated directly through observed data whose value is 18 m. An optimal adjustment was also achieved for the variance, leading to extending the research to other species and comparing the data obtained with other existing distributions.

scholarly journals An inequality for probability density functions arising from a distinguishability problem

1998 ◽  
Vol 39 (3) ◽  
pp. 350-354
Author(s):  
Boris Guljaš ◽  
C. E. M. Pearce ◽  
Josip Pečarić
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Real Line ◽  
Integral Inequality ◽  
Probability Density Functions ◽  
Density Functions ◽  
The Real ◽  
Image Position

AbstractAn integral inequality is established involving a probability density function on the real line and its first two derivatives. This generalizes an earlier result of Sato and Watari. If f denotes the probability density function concerned, the inequality we prove is thatunder the conditions β > α 1 and 1/(β+1) < γ ≤ 1.

scholarly journals Improved Chebyshev inequality: new probability bounds with known supremum of PDF

2018 ◽  
Author(s):  
Tomohiro Nishiyama
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Chebyshev Inequality ◽  
Discrete Probability ◽  
One Dimensional ◽  
Probability Bounds ◽  
Mean And Variance ◽  
Discrete Probability Distributions

In this paper, we derive new probability bounds for Chebyshev's inequality if the supremum of the probability density function is known.This result holds for one-dimensional or multivariate continuous probability distributions with finite mean and variance (covariance matrix).We also show that the similar result holds for specific discrete probability distributions.

scholarly journals Estimacion de los valores de retorno de la altura de ola en la Provincia de Buenos Aires, Argentina

1997 ◽  
Vol 24 (1-2) ◽  
pp. 13
Author(s):  
FERNANDO CAVIGLIA ◽  
JORGE POUSA
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Wave Height ◽  
Standard Method ◽  
Buenos Aires ◽  
Extreme Values ◽  
Probability Distributions ◽  
Best Fit ◽  
Return Value

The wave height recorded at 8 stations along the oceanic coast of the Province of Buenos Aires, Argentina, was analysed to estimate the 50-year return value of wave height at each station. The probability distributions of wave height for the measurements made at Mar de Ajó (1976/85), punta Médanos (northern and southern waveriders, 1981/84), Pinamar (1976/91), Mar del Plata (1968/69), Puerto Quequén (1975/76), Pehuen-Có (1986/88) and El Cóndor (1988) summer resort were tested with the theoretical distributions of Rayleigh, Weibull and Fisher-Tippett I. Excluding Mar de Ajó and El Cóndor, the best fit was obtained with the Fisher-Tippett probability density function. The method of Battjes for estimating the return values of wave height was applied and the resulting 50-year return values were 2.80; 6.90; 7.90; 7.20; 7.21; 8.20; 4.30 and 2.84 m for Mar de Ajó, punta Médanos (northern waverider), punta Médanos (southern waverider), Pinamar, Mar del Plata, Puerto Quequén, Pehuen-Có and El Cóndor, respectively. Lastly, the standard method of extreme values was used to analyse 10 and 16 annual wave height maxima from Mar de Ajó and Pinamar, respectively. The 50-year return values were found to be 2.30 m for Mar de Ajó and 7.20 m for Piramar.

On Random Variables which have the same Distribution as their Reciprocals

1965 ◽  
Vol 8 (6) ◽  
pp. 819-824 ◽  
Author(s):  
V. Seshadri
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Interesting Property ◽  
Distribution Law ◽  
Remarkable Property ◽  
F Distribution

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

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