CONSTANCY OF REGRESSION OF A POLYNOMIAL OF SAMPLE AVERAGE OF POSITIVE RANDOM VARIABLES ON THEIR RATIOS CHARACTERIZES GAMMA DISTRIBUTION

1989 ◽  
pp. 309-316
Author(s):  
Abram KAGAN
Keyword(s):  
Gamma Distribution ◽  
Random Variables ◽  
Sample Average ◽  
Constancy Of Regression

Characterization of a class of second-order density functions

1967 ◽  
Vol 4 (1) ◽  
pp. 123-129 ◽  
Author(s):  
C. B. Mehr
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Gaussian Distribution ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Density Functions ◽  
Linear Filtering ◽  
Non Linear

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

Replacement Policies for an Independent Damage Process

2020 ◽  
Vol 28 (01) ◽  
pp. 2150005
Author(s):  
Satoshi Mizutani ◽  
Xufeng Zhao ◽  
Toshio Nakagawa
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Random Variables ◽  
Expected Cost ◽  
Numerical Examples ◽  
Preventive Replacement ◽  
Damage Process ◽  
Random Times ◽  
Optimal Formula

This paper discusses preventive replacement policies for an independent damage process, in which successive shocks occur at random times, and the independent damages caused by shocks are random variables. It is assumed that a unit fails when the damage has exceeded a prespecified level [Formula: see text]. We consider three models: Model 1: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text], or at time [Formula: see text], whichever occurs first. Model 2: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text] or at shock [Formula: see text], whichever occurs first. Model 3: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text], at shock [Formula: see text] or at time [Formula: see text], whichever occurs first. We obtain the expected cost rates for each model and discuss analytical optimal [Formula: see text] and [Formula: see text] to minimize their expected cost rates. Numerical examples are given when the damage has an exponential distribution and shocks occur at a gamma distribution.

scholarly journals Some estimates of the normal approximation for mixture of Poisson and gamma random variables

2010 ◽  
Vol 51 ◽  
Author(s):  
Jonas Kazys Sunklodas
Keyword(s):  
Poisson Distribution ◽  
Gamma Distribution ◽  
Upper Bound ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Lp Norms

In the paper, we present the upper bound of Lp norms ∆p of the order (a1 + a2)/(DZ)-1/2 for all 1 < p< ∞, of the normal approximation for a standardized random variable (Z - EZ)/√DZ, where the random variable Z = a1X + a2Y , a1 + a2 = 1, ai > 0, i = 1, 2, the random variable X is distributed by the Poisson distribution with the parameter λ > 0, and the random variable Y by the standard gamma distribution Γ (α, 0, 1) with the parameter α > 0.

scholarly journals ESTIMATION OF THE NUMBER OF SUMMANDS OF THE CENTRAL LIMIT THEOREM

2020 ◽  
pp. 7-19
Author(s):  
Antonina Ganicheva ◽  
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
Random Variables ◽  
Independent Random Variables ◽  
Distribution Law ◽  
The Central Limit Theorem ◽  
Sample Average ◽  
Normal Probability Distribution

The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.

A characterization of the gamma distribution from a random difference equation

1988 ◽  
Vol 25 (1) ◽  
pp. 142-149 ◽  
Author(s):  
Eric S. Tollar
Keyword(s):  
Difference Equation ◽  
Gamma Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Random Difference Equation ◽  
Random Difference

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

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