Memory As Captured By Distribution Functions

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 10/19/04

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    Date: Tue, 19 Oct 2004 17:53:04 +0000 (UTC)

     From Osher Doctorow mdoctorow@comcast.net

    COPYRIGHT NOTICE
    Memory As Captured By Distribution Functions
    Copyright By Owner Osher Doctorow Ph.D.
    First Published 2004.

    How do we move from two-time-period-back memory, as in Fibonacci
    sequences or numbers, to "all past time memory"?

    The simplest way is by the cumulative distribution function FX(x)
    of random variable X, which is:

    1) FX(x) = P{w: X(w) < = x}.

    The elements of the set { } in braces are the "all past time memory"
    at least for the variable X if all the values of X (x, etc.) are
    time values. I will restrict myself to such cases here.

    Similarly, the bivariate cdf F(x,y) is defined as:

    2) F(x,y) = P{{w, v): X(w) < = x, Y(v) < = y}

    Now consider the equation:

    3) dFX(x)/dx = fX(x) = A + BFX(x) + CFX(x)^2

    Leting x --> infinity yields:

    4) 0 = A + B + C

    since FX(x) --> 1 as x --> infinity. Letting x --> -infinity (or
    0 if X is nonnegative valued) yields:

    5) 0 = A + 0 + 0

    since FX(x) --> 0 as x --> -infiniy. Finally, substituting from
    (4) and (5) into (3) yields:

    6) fX(x) = BFX(x) - BFX(x)^2

    with B > 0. This has the form:

    7) fX(x) = BFX(x)(1 - FX(x))

    which has logistic form for the right hand side.

    Osher Doctorow

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