If an object is moving on the x-axis (straight line) between 0
(starting point) and X (end point)at a variable speed x(t)and
eventually may or may not reach x(t)=X (end point) at time T (time to
complete the trip) depending on the function x(t).

Question : 
Only in terms of function x(t), derive a general probability function
P[x(t)]for finding the object at an exact position(x) at any given
time t <= T.

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Clarification of Question by sabawi-ga on 22 Jan 2005 08:12 PST

Note: that x(t) is the integral of the velocity function dx/dt =
v(x,t) and not the actual velocity function. The x(t) function returns
the distance x from the origin 0 after time value t has passed.

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Clarification of Question by sabawi-ga on 22 Jan 2005 14:36 PST

Assume random fluctuations in the frinction ? along the movement path
of the object as a function of velicity (i.e. dx/dt = v(x,t) + v(x,t)
?(x,t))

It is difficult for me to envision this as a probability problem, at
least without further assumptions about "x(t)".

Assuming the position function x(t) is given as continuous, the set of
points visited during the interval [0,T] is simply:

  S(T) :=  [ min(x[0,T]),max(x[0,T]) ]

where x[0,T] is an abbreviation for {x(t) : 0 <= t <= T}.

I suppose that one can describe the "probability" as 1 for x in S(T)
and 0 for x outside of S(T), but this seems a trivial observation.

Presumably there are some additional assumptions to be made about x(t)
which contribute to an interpretation in terms of probability.  For
example, we might  "replace" x(t) by a random process X(t) and specify
that X(t) satisfies a stochastic differential equation.  Such problems
arise in portfolio management and other "financial analytics"
applications, where one wishes to estimate the likelihood of reaching
an investment object within a "horizon" of time T.

See for example:

[Brownian motion -- Wikipedia]
http://en.wikipedia.org/wiki/Brownian_motion


regards, mathtalk-ga

Ditto on what mathtalk has written.

In addition, you should be aware that for any continuous probability
distribution function (that is, excluding "pathological" cases such as
when the pdf is composed of a sum of delta functions), the probability
of finding the object at an *exact* position, x, is exactly and
precisely zero.  Continuous distributions don't work like discrete
probability distributions (see, for example
<http://www.mathworks.com/access/helpdesk/help/toolbox/stats/prob_di7.html>,
or any basic probability text).  What you *can* ask (and answer, if
you have the distribution) is "what is the probability that x lies
between the two values X1 and X2".