exponential distribution

Then, to illustrate the applicability of the proposed new family, it is considered the particular case of the distribution obtained when taking into account that G (x) is the distribution function of an exponential random variable. By presenting mathematical structures for gamma- [(1 - G)/G] class, it was also derived statistical properties from this new distribution, and, to illustrate its potentiality, an application to a set of real data is performed.

where the first and second equalities hold due to the order statistics [29] of an exponential random variable and the binomial theorem, respectively.

Theorem 1: If [U.sub.1] is an exponential random variable with E {[U.sub.l]} = [[lambda].sub.1], [U.sub.2] is a sum of N statistically independent exponential random variables, that is [U.sub.2] = [N.summation over (j=1)], [Y.sub.j] with E {[Y.sub.j]} = [[mu].sub.j], then the cumulative distribution function (c.d.f.) of the random variable U = [U.sub.1] / 1 + [U.sub.2] can be obtained as

realizations of standard exponential random variable vector z in (7) are generated numerically, while the eigenvalue estimates are calculated using both estimators with the number of channel samples N drawn from the set of {10, 20, 50}.

P3: Since a standard exponential random variable has the same distribution as log(U) where U is the standard uniform variable, we can represent X in terms of two i.i.d standard uniform variables [U.sub.1] and [U.sub.2] as follows.

[[xi].sup.1].sub.1] + [[xi].sup.0].sub.1] = 0 The time in state 0 is an exponential random variable with mean,1 [[xi].sup.0].sub.1] Also, the time in state 1 is an exponential random variable, with mean,1/[[xi].sup.0].sub.1].