By Samuel Kotz

ISBN-10: 146120173X

ISBN-13: 9781461201731

ISBN-10: 1461266467

ISBN-13: 9781461266464

The target of this monograph is kind of modest: It makes an attempt to be a scientific exposition of all that seemed within the literature and was once recognized to us via the tip of the 20 th century in regards to the Laplace distribution and its various generalizations and extensions. we've attempted to hide either theoretical advancements and functions. there have been major purposes for penning this e-book. the 1st was once our conviction that the parts and occasions the place the Laplace distribution certainly happens is so huge that monitoring the unique resources is unfeasible. the second one was once our commentary of the becoming call for for statistical distributions having houses tangent to these exhibited by means of the Laplace legislation. those "necessary" stipulations justified our efforts that ended in this e-book. Many information are prepared basically for reference, resembling inclusion of the main general terminology and notation. in different situations, now we have proposed unification to beat the anomaly of notions so frequently found in this quarter. own flavor could have performed a few injustice to the subject material by means of omitting or emphasizing convinced issues because of area obstacles. We belief that this option doesn't represent a major drawback-in our literature seek we attempted to go away no stone unturned (we accrued over four hundred references).

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**Extra resources for The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance**

**Sample text**

42 2. 4 . I I· . d. standard Laplace random variables (dashed lines) vs. d. standard Gaussian random variables (solid lines). d. case the densities of the ratio of two mean-zero Laplace and two mean-zero normal variables are 1( 2 1)2 1 1 1 + Ix I and -; 1 + x 2 ' x E JR, respectively. 3 (bottom). 3. 48). 5 below summarizes our results on distributions of common functions of independent Laplace variables. Distribution Function YI + ... 2n x E lR. j! -2-j-' 1,JSIs2(l 1~ I f(x) -- (n-I+j)! Ixl n- 1- I:n-I (se-s2Ixl_s2e-sllxl) I ~K SP2 0 (2{fff) spz' l~ 2 SI ( I l+(sz/sl)lxl f Sl = S2, , Sl f= S2, x E lR.

S Wi. We thus must find the distribution of the ratio of two independent standard exponential random variables. First, we find the distribution function by conditioning. For x > 0 we have -:s x ) = P ( WI W2 1 00 P(WI :s xz)e-zdz = 1- 0 1° 00 e-z(x+l)dz = 1 - -1 . , Johnson et al. (1994), p. 575, or Springer (1979), p. 161] with density f W l/ W2(X) = C~xy, x:::: o. 38) has infinite mean and variance. (However, 12It should be noted that our result does not fully agree with Weida (1935). 42 2. 4 .

Indeed, since for each i = 1, ... d. d. 7) with the shape parameter v = n. 9) with v = nand K = 1. 2n (n _ 1 ~ (n _ 1=0 + j)! j! 1- j 2j , x E JR.. 18). 2 Let Xl, ... d. standard classical Laplace variables. Then Mn T = Xl + ... + Xn 1: J. 27) j=l where the Z j 's are i. d. v. given by the probability function . 2j P(Mn =J)=2 2n - 1 (2n - j n-l [The Zj'S, J, and Mn are mutually independent, and 1) ' j = 1,2, ... , n. 4 below contains the densities of X n for sample sizes n = 1, 2, 3, 4, which were worked out in Craig (1932)11 [see also Edwards (1948)].

### The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance by Samuel Kotz

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