A Simple Modification of the Binomial Distribution

1960 ◽  
Vol 15 (06) ◽  
pp. 436-444
Author(s):  
S. W. Dharmadhikari
Keyword(s):  
Probability Distribution ◽  
Poisson Distribution ◽  
Binomial Distribution ◽  
Probability Distributions ◽  
Simple Modification ◽  
Type Iii

Given any probability distribution, new distributions can be derived from it by assuming its parameters to follow some specific probability distributions. A simple example of this process is provided by the Poisson distributionP(r∣λ) =e-λλr/r! (r= o, 1, 2, …).If the parameterλis assumed to follow the Pearson's Type III lawthen the probability ofrsuccesses is obtained as

Probability and distributions

2020 ◽  
pp. 243-280
Author(s):  
Janet L. Peacock ◽  
Philip J. Peacock
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Poisson Distribution ◽  
Binomial Distribution ◽  
Probability Distributions ◽  
Medical Statistics

Probability and probability distributions play a central part in medical statistics. This chapter defines what is meant by probability and describes the rules by which probabilities are combined. It then describes how the use of probability leads to the concept of a probability distribution and shows how these distributions are used in medical statistics. Examples are given of the use of key distributions: the Normal distribution, the binomial distribution, and the Poisson distribution.

scholarly journals A best-fit probability distribution for the estimation of rainfall in northern regions of Pakistan

2016 ◽  
Vol 11 (1) ◽  
pp. 432-440 ◽  
Author(s):  
M. T. Amin ◽  
M. Rizwan ◽  
A. A. Alazba
Keyword(s):  
Probability Distribution ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Future Research ◽  
Maximum Rainfall ◽  
Type Iii ◽  
Goodness Of Fit Tests ◽  
Pearson Type ◽  
Northern Regions ◽  
Best Fit

AbstractThis study was designed to find the best-fit probability distribution of annual maximum rainfall based on a twenty-four-hour sample in the northern regions of Pakistan using four probability distributions: normal, log-normal, log-Pearson type-III and Gumbel max. Based on the scores of goodness of fit tests, the normal distribution was found to be the best-fit probability distribution at the Mardan rainfall gauging station. The log-Pearson type-III distribution was found to be the best-fit probability distribution at the rest of the rainfall gauging stations. The maximum values of expected rainfall were calculated using the best-fit probability distributions and can be used by design engineers in future research.

scholarly journals Theorizing Probability Distribution in Applied Statistics

2020 ◽  
Vol 1 (1) ◽  
pp. 79-95
Author(s):  
Indra Malakar
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Poisson Distribution ◽  
Binomial Distribution ◽  
Identical Condition ◽  
Random Variable ◽  
Fixed Number ◽  
Theoretical Knowledge ◽  
Applied Statistics ◽  
Discrete Random Variable

This paper investigates into theoretical knowledge on probability distribution and the application of binomial, poisson and normal distribution. Binomial distribution is widely used discrete random variable when the trails are repeated under identical condition for fixed number of times and when there are only two possible outcomes whereas poisson distribution is for discrete random variable for which the probability of occurrence of an event is small and the total number of possible cases is very large and normal distribution is limiting form of binomial distribution and used when the number of cases is infinitely large and probabilities of success and failure is almost equal.

scholarly journals A mechanistic model for the negative binomial distribution of single-cell mRNA counts

2019 ◽  
Author(s):  
Lisa Amrhein ◽  
Kumar Harsha ◽  
Christiane Fuchs
Keyword(s):  
Steady State ◽  
Probability Distribution ◽  
Single Cell ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Mechanistic Model ◽  
Steady State Probability ◽  
Degradation Models

SummarySeveral tools analyze the outcome of single-cell RNA-seq experiments, and they often assume a probability distribution for the observed sequencing counts. It is an open question of which is the most appropriate discrete distribution, not only in terms of model estimation, but also regarding interpretability, complexity and biological plausibility of inherent assumptions. To address the question of interpretability, we investigate mechanistic transcription and degradation models underlying commonly used discrete probability distributions. Known bottom-up approaches infer steady-state probability distributions such as Poisson or Poisson-beta distributions from different underlying transcription-degradation models. By turning this procedure upside down, we show how to infer a corresponding biological model from a given probability distribution, here the negative binomial distribution. Realistic mechanistic models underlying this distributional assumption are unknown so far. Our results indicate that the negative binomial distribution arises as steady-state distribution from a mechanistic model that produces mRNA molecules in bursts. We empirically show that it provides a convenient trade-off between computational complexity and biological simplicity.Graphical Abstract

On a class of probability distributions

1949 ◽  
Vol 45 (2) ◽  
pp. 219-224 ◽  
Author(s):  
S. Chandrasekhar
Keyword(s):  
Probability Distribution ◽  
Explicit Formula ◽  
Poisson Distribution ◽  
Probability Distributions ◽  
Space Density ◽  
Special Cases ◽  
Distribution Of Points ◽  
Image Position ◽  
Infinite Sum ◽  
Finite Moments

1. The statement of the problem. A problem which arises in certain physical (5) and astronomical (1, 3) contexts relates to the probability distribution of the infinite sumwhere the rj's are the position vectors from a fixed origin of an infinite uniform Poisson distribution of points with a given constant mean space density N. While certain special cases of the sum (1) have been considered in the literature (cf. (2)) the study of the general case discloses properties of a class of probability distributions which appear to have some interest. For example, we shall show that the probability distribution of F exists only for that the moments of F (= |F|) exist only for orders 0 ≤ p < 3/n, and finally that an explicit formula can be given for all the finite moments.

scholarly journals Comparative Evaluation of Probability Distribution Models of Flood flow in Lower Niger Basin

2021 ◽  
Vol 6 (2) ◽  
pp. 107-117
Author(s):  
Itolima Ologhadien
Keyword(s):  
Water Resources ◽  
Probability Distribution ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Distribution Model ◽  
Distribution Models ◽  
Optimum Distribution ◽  
Water Resources Systems ◽  
Type Iii ◽  
Pearson Type

The choice of optimum probability distribution model that would accurately simulate flood discharges at a particular location or region has remained a challenging problem to water resources engineers. In practice, several probability distributions are evaluated, and the optimum distribution is then used to establish the quantile - probability relationship for planning, design and management of water resources systems, risk assessment in flood plains and flood insurance. This paper presents the evaluation of five probability distributions models: Gumbel (EV1), 2-parameter lognormal (LN2), log pearson type III (LP3), Pearson type III(PR3), and Generalised Extreme Value (GEV) using the method of moments (MoM) for parameter estimation and annual maximum series of five hydrological stations in the lower Niger River Basin in Nigeria. The choice of optimum probability distribution model was made on five statistical goodness – of – fit measures; modified index of agreement (Dmod), relative root mean square error (RRMSE), Nash – Sutcliffe efficiency (NSE), Percent bias (PBIAS), ratio of RMSE and standard deviation of the measurement (RSR), and probability plot correlation coefficient (PPCC). The results show that GEV is the optimum distribution in 3 stations, and LP3 in 2 stations. On the overall GEV is the best – fit distribution, seconded by PR3 and thirdly, LP3. Furthermore, GEV simulated discharges were in closest agreement with the observed flood discharges. It is recommended that GEV, PR3 and LP3 should be considered in the final selection of optimum probability distribution model in Nigeria.

Probability and distributions

2010 ◽  
pp. 203-235
Author(s):  
Janet L. Peacock ◽  
Philip J. Peacock
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Poisson Distribution ◽  
Central Limit ◽  
Binomial Distribution ◽  
Probability Distributions ◽  
Percentage Points

Independence: data and variables 204 Probability: definitions 206 Probability: properties 208 Probability distributions 210 Binomial distribution: formula 212 Binomial distribution: derivation 214 Poisson distribution 216 Continuous probability distributions 220 Normal distribution 222 Normal distribution: calculating probabilities 224 Normal distribution: percentage points 226 Central limit theorem ...

WIGNER FUNCTION OF PULSED FIELDS RECONSTRUCTED BY DIRECT DETECTION

2011 ◽  
Vol 09 (supp01) ◽  
pp. 39-47
Author(s):  
ALESSIA ALLEVI ◽  
MARIA BONDANI ◽  
ALESSANDRA ANDREONI
Keyword(s):  
Probability Distribution ◽  
Wigner Function ◽  
Beam Splitter ◽  
Direct Detection ◽  
Probability Distributions ◽  
Linear Regime ◽  
Wigner Functions ◽  
Intensity Measurements ◽  
Pulsed Fields

We present the experimental reconstruction of the Wigner function of some optical states. The method is based on direct intensity measurements by non-ideal photodetectors operated in the linear regime. The signal state is mixed at a beam-splitter with a set of coherent probes of known complex amplitudes and the probability distribution of the detected photons is measured. The Wigner function is given by a suitable sum of these probability distributions measured for different values of the probe. For comparison, the same data are analyzed to obtain the number distributions and the Wigner functions for photons.

Analysis of Magnitude and Frequency of Floods in the Damanganga Basin: Western India

2021 ◽  
Vol 5 (1) ◽  
pp. 1-11
Author(s):  
Vitthal Anwat ◽  
Pramodkumar Hire ◽  
Uttam Pawar ◽  
Rajendra Gunjal
Keyword(s):  
Probability Distribution ◽  
Probability Distributions ◽  
Flood Frequency ◽  
Flood Frequency Analysis ◽  
Western India ◽  
Type I ◽  
Return Periods ◽  
Pearson Type ◽  
Kolmogorov Smirnov ◽  
Anderson Darling

Flood Frequency Analysis (FFA) method was introduced by Fuller in 1914 to understand the magnitude and frequency of floods. The present study is carried out using the two most widely accepted probability distributions for FFA in the world namely, Gumbel Extreme Value type I (GEVI) and Log Pearson type III (LP-III). The Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) methods were used to select the most suitable probability distribution at sites in the Damanganga Basin. Moreover, discharges were estimated for various return periods using GEVI and LP-III. The recurrence interval of the largest peak flood on record (Qmax) is 107 years (at Nanipalsan) and 146 years (at Ozarkhed) as per LP-III. Flood Frequency Curves (FFC) specifies that LP-III is the best-fitted probability distribution for FFA of the Damanganga Basin. Therefore, estimated discharges and return periods by LP-III probability distribution are more reliable and can be used for designing hydraulic structures.

Analysis and Synthesis of Mechanical Error in Universal Joints

1990 ◽  
Author(s):  
J. L. Cagney ◽  
S. S. Rao
Keyword(s):  
Probability Distribution ◽  
Real World ◽  
Probability Distributions ◽  
Manufacturing Cost ◽  
Universal Joint ◽  
Accuracy Requirement ◽  
Output Error ◽  
Manufacturing Errors ◽  
Analysis And Synthesis ◽  
Limiting Value

Abstract The modeling of manufacturing errors in mechanisms is a significant task to validate practical designs. The use of probability distributions for errors can simulate manufacturing variations and real world operations. This paper presents the mechanical error analysis of universal joint drivelines. Each error is simulated using a probability distribution, i.e., a design of the mechanism is created by assigning random values to the errors. Each design is then evaluated by comparing the output error with a limiting value and the reliability of the universal joint is estimated. For this, the design is considered a failure whenever the output error exceeds the specified limit. In addition, the problem of synthesis, which involves the allocation of tolerances (errors) for minimum manufacturing cost without violating a specified accuracy requirement of the output, is also considered. Three probability distributions — normal, Weibull and beta distributions — were used to simulate the random values of the errors. The similarity of the results given by the three distributions suggests that the use of normal distribution would be acceptable for modeling the tolerances in most cases.

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