On a characterization of the exponential distribution by order statistics

1976 ◽  
Vol 13 (4) ◽  
pp. 818-822 ◽  
Author(s):  
M. Ahsanullah
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Order Statistics ◽  
Density Function ◽  
Exponential Distribution ◽  
Random Sample ◽  
Random Variables ◽  
Identical Distribution

Let X1, X2, …, Xn be a random sample of size n from a population with probability density function f(x), x >0, and let X1,n < X2,n < … < Xn,n be the associated order statistics. A characterization of the exponential distribution is shown by considering the identical distribution of the random variables nX1,n and (n − i + 1)(X1,n −; Xi–1,n) for one i and one n with 2 ≦ i ≦ n.

A characterization of the exponential distribution by spacings

1978 ◽  
Vol 15 (3) ◽  
pp. 650-653 ◽  
Author(s):  
M. Ahsanullah
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Order Statistics ◽  
Density Function ◽  
Exponential Distribution ◽  
Random Sample ◽  
Random Variables ◽  
Identical Distribution

Let X1, X2, ···, Xn be a random sample of size n from a population with probability density function f(x), x > 0, and let X1, n < X2, n < ··· < Xn, n be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − Xi−1, n) and (n − i)(Xi+1, n − Xi, n) for one i and one n with 2 ≦ i ˂ n.

A characterization of the exponential distribution by spacings

1978 ◽  
Vol 15 (03) ◽  
pp. 650-653 ◽  
Author(s):  
M. Ahsanullah
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Order Statistics ◽  
Density Function ◽  
Exponential Distribution ◽  
Random Sample ◽  
Random Variables ◽  
Identical Distribution

Let X 1, X 2, ···, Xn be a random sample of size n from a population with probability density function f(x), x &gt; 0, and let X 1, n &lt; X 2, n &lt; ··· &lt; Xn, n be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − X i−1, n ) and (n − i)(X i+1, n − X i, n ) for one i and one n with 2 ≦ i ˂ n.

On a characterization of the exponential distribution by order statistics

1976 ◽  
Vol 13 (04) ◽  
pp. 818-822 ◽  
Author(s):  
M. Ahsanullah
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Order Statistics ◽  
Density Function ◽  
Exponential Distribution ◽  
Random Sample ◽  
Random Variables ◽  
Identical Distribution

Let X 1 , X2, …, Xn be a random sample of size n from a population with probability density function f(x), x &gt;0, and let X 1,n &lt; X 2,n &lt; … &lt; Xn,n be the associated order statistics. A characterization of the exponential distribution is shown by considering the identical distribution of the random variables nX 1,n and (n − i + 1)(X 1,n −; X i–1,n ) for one i and one n with 2 ≦ i ≦ n.

Multivariate probabilistic rock physics models using Kumaraswamy distributions

Geophysics ◽  
2021 ◽  
pp. 1-43
Author(s):  
Dario Grana
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Random Variables ◽  
Rock Physics ◽  
Petrophysical Properties ◽  
Kumaraswamy Distribution ◽  
Model Predictions ◽  
Physics Model ◽  
Rock Physics Model

Rock physics models are physical equations that map petrophysical properties into geophysical variables, such as elastic properties and density. These equations are generally used in quantitative log and seismic interpretation to estimate the properties of interest from measured well logs and seismic data. Such models are generally calibrated using core samples and well log data and result in accurate predictions of the unknown properties. Because the input data are often affected by measurement errors, the model predictions are often uncertain. Instead of applying rock physics models to deterministic measurements, I propose to apply the models to the probability density function of the measurements. This approach has been previously adopted in literature using Gaussian distributions, but for petrophysical properties of porous rocks, such as volumetric fractions of solid and fluid components, the standard probabilistic formulation based on Gaussian assumptions is not applicable due to the bounded nature of the properties, the multimodality, and the non-symmetric behavior. The proposed approach is based on the Kumaraswamy probability density function for continuous random variables, which allows modeling double bounded non-symmetric distributions and is analytically tractable, unlike the Beta or Dirichtlet distributions. I present a probabilistic rock physics model applied to double bounded continuous random variables distributed according to a Kumaraswamy distribution and derive the analytical solution of the posterior distribution of the rock physics model predictions. The method is illustrated for three rock physics models: Raymer’s equation, Dvorkin’s stiff sand model, and Kuster-Toksoz inclusion model.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

Response Probability Analysis of Random Acoustic Field Based on Perturbation Stochastic Method and Change-of-Variable Technique

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Baizhan Xia ◽  
Dejie Yu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Acoustic Field ◽  
Random Variables ◽  
Stochastic Finite Element Method ◽  
Variable Technique ◽  
Change Of Variable

To calculate the probability density function of the response of a random acoustic field, a change-of-variable perturbation stochastic finite element method (CVPSFEM), which integrates the perturbation stochastic finite element method (PSFEM) and the change-of-variable technique in a unified form, is proposed. In the proposed method, the response of a random acoustic field is approximated as a linear function of the random variables based on a first order stochastic perturbation analysis. According to the linear relationship between the response and the random variables, the formal expression of the probability density function of the response of a random acoustic field is obtained by the change-of-variable technique. The numerical examples on a two-dimensional (2D) acoustic tube and a three-dimensional (3D) acoustic cavity of an automobile cabin verify the accuracy and efficiency of the proposed method. Hence, the proposed method can be considered as an effective method to quantify the effects of the parametric randomness of a random acoustic field on the sound pressure response.

Study on Determining Safety-Monitoring Index for Concrete Dam Based on Cloud Model

2012 ◽  
Vol 226-228 ◽  
pp. 1106-1110 ◽  
Author(s):  
Dong Qin ◽  
Xue Qin Zheng ◽  
Song Lin Wang
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Cloud Model ◽  
Safety Monitoring ◽  
Monitoring Data ◽  
Concrete Dam ◽  
Deterministic Function ◽  
Monitoring Index

The paper, based on analyzing original monitoring data, employs forward and backward cloud algorithm in studying determining safety-monitoring index for concrete dam ,which integrates randomness and fuzziness into of qualitative concept of digital features. By means of above monitoring data, its digital characteristics can be easily transformed to the “quantitative-qualitative- quantitative” change. The final generated quantitative value constitutes the cloud diagram where each droplet demonstrates the characterization of raw monitoring data. At the same time, it also shows the randomness and fuzziness of monitored value. we can study out the safety monitoring indexes according to different remarkable levels by using the probability density function and deterministic function which completed by cloud algorithm. In the end, it is obtained with practice that this method is more suitable and reliability.

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