A note on stochastic ordering of order statistics

1997 ◽  
Vol 34 (03) ◽  
pp. 785-789 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Order Statistics ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stochastic Comparisons ◽  
Homogeneous Population ◽  
Independent Components ◽  
Binomial Random Variable ◽  
Necessary And Sufficient

A necessary and sufficient condition is obtained for a Poisson binomial random variable to be stochastically larger (or smaller) than a binomial random variable. It is then used to deal with the stochastic comparisons of order statistics from heterogeneous populations with those from a homogeneous population. The result has obvious applications in the stochastic comparisons of lifetimes of k-out-of-n systems having independent components.

A note on stochastic ordering of order statistics

1997 ◽  
Vol 34 (3) ◽  
pp. 785-789 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Order Statistics ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stochastic Comparisons ◽  
Homogeneous Population ◽  
Independent Components ◽  
Binomial Random Variable ◽  
Necessary And Sufficient

A necessary and sufficient condition is obtained for a Poisson binomial random variable to be stochastically larger (or smaller) than a binomial random variable. It is then used to deal with the stochastic comparisons of order statistics from heterogeneous populations with those from a homogeneous population. The result has obvious applications in the stochastic comparisons of lifetimes of k-out-of-n systems having independent components.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (02) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (2) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

Some Reliability and Other Properties of Beta-Transformed Random Variables

2018 ◽  
Vol 33 (2) ◽  
pp. 83-92
Author(s):  
M. Sreehari ◽  
E. Sandhya ◽  
V. K. Mohamed Akbar
Keyword(s):  
Power Series ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Power Series Distribution ◽  
Geometric Random Variable ◽  
Necessary And Sufficient

Abstract The reliability properties of beta-transformed random variables are discussed. A necessary and sufficient condition for a beta-transformed geometric random variable to follow a power series distribution is derived. It is shown that a beta-transformed member of the Katz family does not belong to the Katz family unless it is a geometric distribution, thereby getting a characterization.

Almost sure convergence in Markov branching processes with infinite mean

1977 ◽  
Vol 14 (4) ◽  
pp. 702-716 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Functional Equation ◽  
Distribution Function ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

scholarly journals Entrance Fees and a Bayesian Approach to the St. Petersburg Paradox

2017 ◽  
Author(s):  
Diego Marcondes ◽  
Cláudia Peixoto ◽  
Kdson Souza ◽  
Sergio Wechsler
Keyword(s):  
Bayesian Approach ◽  
Random Variable ◽  
Natural Generalization ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Fair Coin ◽  
Expected Gain ◽  
Necessary And Sufficient ◽  
Petersburg Paradox ◽  
The Bayesian Approach

In his best-selling book An Introduction to Probability Theory and its Applications, W. Feller established a way of ending the St. Petersburg Paradox by the introduction of an entrance fee, and provided it for the case in which the game is played with a fair coin. A natural generalization of his method is to establish the entrance fee for the case in which the probability of head is θ (0 < θ < 1/2). The deduction of those fees is the main result of Section 2. We then propose a Bayesian approach to the problem. When the probability of head is θ (1/2 < θ < 1) the expected gain of the St. Petersburg Game is finite, therefore there is no paradox. However, if one takes θ as a random variable assuming values in (1/2,1) the paradox may hold, what is counter-intuitive. On Section 3 we determine a necessary and sufficient condition for the absence of paradox on the Bayesian approach and on Section 4 we establish the entrance fee for the case in which θ is uniformly distributed in (1/2,1), for in this case there is paradox.

Almost sure convergence in Markov branching processes with infinite mean

1977 ◽  
Vol 14 (04) ◽  
pp. 702-716 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Functional Equation ◽  
Distribution Function ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

If {Zn } is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn } and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt } with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

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