Identically distributed linear forms and the normal distribution

1973 ◽  
Vol 5 (01) ◽  
pp. 138-152
Author(s):  
S. G. Ghurye ◽  
I. Olkin
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Linear Forms ◽  
Multivariate Case ◽  
Identical Distribution

A general discussion and survey is provided of the characterization of the normal distribution by the identical distribution of linear forms. The first result dates to 1923 when Pólya showed that if X and Y are i.i.d. random variables satisfying certain conditions, and if aX + bY is distributed as (a 2 + b 2)1/2 X, then X has a normal distribution. This result has been generalized in several directions. In addition to a recasting of some of the results, an extension in the multivariate case is provided.

Identically distributed linear forms and the normal distribution

1973 ◽  
Vol 5 (1) ◽  
pp. 138-152 ◽  
Author(s):  
S. G. Ghurye ◽  
I. Olkin
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Linear Forms ◽  
Multivariate Case ◽  
Identical Distribution

A general discussion and survey is provided of the characterization of the normal distribution by the identical distribution of linear forms. The first result dates to 1923 when Pólya showed that if X and Y are i.i.d. random variables satisfying certain conditions, and if aX + bY is distributed as (a2 + b2)1/2X, then X has a normal distribution. This result has been generalized in several directions. In addition to a recasting of some of the results, an extension in the multivariate case is provided.

Characterization of distributions by the identical distribution of linear forms

1966 ◽  
Vol 3 (02) ◽  
pp. 481-494 ◽  
Author(s):  
Morris L. Eaton
Keyword(s):  
Random Variables ◽  
Linear Forms ◽  
Characterization Of Distributions ◽  
Identical Distribution ◽  
The Law

Throughout this paper, we shall write ℒ(W) = ℒ(Z) to mean the random variables W and Z have the same distribution. The relation “ℒ(W) = ℒ(;Z)” reads “the law of W equals the law of Z”.

Characterization of distributions by the identical distribution of linear forms

1966 ◽  
Vol 3 (2) ◽  
pp. 481-494 ◽  
Author(s):  
Morris L. Eaton
Keyword(s):  
Random Variables ◽  
Linear Forms ◽  
Characterization Of Distributions ◽  
Identical Distribution ◽  
The Law

Throughout this paper, we shall write ℒ(W) = ℒ(Z) to mean the random variables W and Z have the same distribution. The relation “ℒ(W) = ℒ(;Z)” reads “the law of W equals the law of Z”.

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.

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