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In article <[EMAIL PROTECTED]>,
ZHANG Yan <[EMAIL PROTECTED]> wrote:
>Suppose that X is nonnegative continuous random variable with
>probability density function f_X(x).
>Now, we have
>A = Integral ( InverseLaplace(g(s)) f_X(x), 0, INFINITE)
>where Integral( ..., 0, INFINITE) represents the integral from zero to
>positive infinite. InverseLaplace(g(s)) represents the inverse
>laplace of g(s). We can denote that
>G(x) = InverseLaplace(g(s)).
>My question is as follows:
>If A is greater(or less) than zero, then what condition should the
>function g(s) satisfy, or any requirement for the function g(s)?
I assume you mean A >= 0 for all such f. I'll also ignore the fact that
in nontrivial cases there are some f for which the integral diverges.
By taking suitable limits ("approximate delta functions"), you must
have G(x) >= 0 on [0,infinity). So the condition is that g(s) is the
Laplace transform of a nonnegative function.
Robert Israel [EMAIL PROTECTED]
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
