Chapter 14. Continuous Random Variables: The Normal Distribution

2017 ◽  
pp. 371-397
Keyword(s):  
Normal Distribution ◽  
Random Variables

On the Mathematical Basis of Zelen’s Prerandomized Designs

1985 ◽  
Vol 24 (03) ◽  
pp. 120-130 ◽  
Author(s):  
E. Brunner ◽  
N. Neumann
Keyword(s):  
Clinical Trial ◽  
Normal Distribution ◽  
Random Variables ◽  
Small Samples ◽  
Selection Effects ◽  
Sample Sizes ◽  
Mathematical Basis ◽  
Additional Assumptions

SummaryThe mathematical basis of Zelen’s suggestion [4] of pre randomizing patients in a clinical trial and then asking them for their consent is investigated. The first problem is to estimate the therapy and selection effects. In the simple prerandomized design (PRD) this is possible without any problems. Similar observations have been made by Anbar [1] and McHugh [3]. However, for the double PRD additional assumptions are needed in order to render therapy and selection effects estimable. The second problem is to determine the distribution of the statistics. It has to be taken into consideration that the sample sizes are random variables in the PRDs. This is why the distribution of the statistics can only be determined asymptotically, even under the assumption of normal distribution. The behaviour of the statistics for small samples is investigated by means of simulations, where the statistics considered in the present paper are compared with the statistics suggested by Ihm [2]. It turns out that the statistics suggested in [2] may lead to anticonservative decisions, whereas the “canonical statistics” suggested by Zelen [4] and considered in the present paper keep the level quite well or may lead to slightly conservative decisions, if there are considerable selection effects.

scholarly journals A refinement of normal approximation to Poisson binomial

2005 ◽  
Vol 2005 (5) ◽  
pp. 717-728 ◽  
Author(s):  
K. Neammanee
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Bernoulli Random Variables ◽  
Binomial Random Variable ◽  
Standard Normal ◽  
Correction Terms ◽  
Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

Error Oblate

2010 ◽  
Vol 40-41 ◽  
pp. 941-947
Author(s):  
Yang Sheng You ◽  
Yang Hua Gao ◽  
Yan Ying Chen ◽  
Xin Jie Deng
Keyword(s):  
Data Processing ◽  
Normal Distribution ◽  
Natural Extension ◽  
Random Variables ◽  
Error Ellipse ◽  
Theoretical Significance ◽  
Concept Theory

Error Ellipse can’t correctly depict the distribution of random variables in non-normal distribution, so this paper presents the concept, theory and methods of Error Oblate. Error Oblate is the natural extension of Error Ellipse. It is widely applicable and carries great theoretical significance in data processing.

scholarly journals Tail Properties and Asymptotic Expansions for the Maximum of the Logarithmic Skew-Normal Distribution

2013 ◽  
Vol 50 (3) ◽  
pp. 900-907 ◽  
Author(s):  
Xin Liao ◽  
Zuoxiang Peng ◽  
Saralees Nadarajah
Keyword(s):  
Asymptotic Expansion ◽  
Convergence Rate ◽  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Skew Normal Distribution ◽  
Skew Normal

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

scholarly journals Rigorous upper bounds on data complexities of block cipher cryptanalysis

2017 ◽  
Vol 11 (3) ◽  
Author(s):  
Subhabrata Samajder ◽  
Palash Sarkar
Keyword(s):  
Statistical Analysis ◽  
Normal Distribution ◽  
Block Cipher ◽  
Random Variables ◽  
Upper Bounds ◽  
Statistical Analyses ◽  
Data Complexity ◽  
Sum Of Random Variables ◽  
Symmetric Key

AbstractStatistical analysis of symmetric key attacks aims to obtain an expression for the data complexity which is the number of plaintext-ciphertext pairs needed to achieve the parameters of the attack. Existing statistical analyses invariably use some kind of approximation, the most common being the approximation of the distribution of a sum of random variables by a normal distribution. Such an approach leads to expressions for data complexities which are

Arithmetic Research of Computer Simulation of the Bus Junction Stations

2010 ◽  
Vol 108-111 ◽  
pp. 441-445
Author(s):  
Yong Luo ◽  
Xiu Chun Guo
Keyword(s):  
Computer Simulation ◽  
Normal Distribution ◽  
Exponential Distribution ◽  
Poisson Distribution ◽  
Computer Model ◽  
Time Distribution ◽  
Random Variables ◽  
Step Length ◽  
Negative Exponential Distribution ◽  
Negative Exponential

This paper will construct a discrete system's computer model, include the time distribution of buses arrive, stop and passengers’ get down and get off. The emulation clock advanced with the method of incident step length. Through the inversion produce the random variables of Poisson distribution, negative exponential distribution and normal distribution, simulate the conditions of bus arrive, leave and pick up passengers. Finally calculate with the relative data and then get a series of indicators to evaluate the crowing degree of the bus station.

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