Date: Feb 10, 2009 8:52 PM
Subject: Re: function of uniform probability

On Feb 10, 3:21 pm, Boris Gourévitch <>
> Hi,

Hi, I'm not entirely sure I understand your question, so if my reply
doesn't make any sense then please ignore it.

> I'm working in signal processing and I would like to build a function
> f(t) (t is time) defined on R+, bounded on [a,b], derivable

I'm wondering if you might mean "differentiable" rather than

> and which
> verifies Probability(f(t) is in [c,d])=(d-c), [c,d] being included in
> [a,b].

Do you mean f(t) at a uniformly randomly chosen instant in time? I'm
not sure it's logically possible to choose such an instant unless you
constrain t to lie in a finite time interval -- though you can
sidestep this issue by using a periodic function. Also, probability is
a number between zero and one, so I'm wondering if you mean the
probability is *proportional* to d - c, rather than equal to it.

I'm guessing that your probability requirements are such that a
sawtooth wave would work, except that's no good to you because it's
not smooth. I don't see that a smooth bounded periodic function
defined for all t > 0 is possible: f(t) will inevitably be biased
towards the points of maximum and minimum when arbitrarily small
intervals [c,d] are considered. For the same reason, I don't see that
a smooth function defined over a finite timespan (with t chosen
uniformly randomly from that timespan) will be possible either, except
for the trivial case of a linear function. The only way I can see is
to somehow contrive to have a maximum or minimum at every value within
[a,b] (for some t) and make the limiting probability over t = 0 to t =
T come right as T -> infinity, if that's possible (I don't know if it
is). But the probabilities wouldn't be right with t chosen from any
finite time interval.

> in other words, I would like a function that meets each point of
> [a,b] with the same probability, If I express it correctly.
> An additional constraint on f would be that its frequential spectrum
> (with a given sampling frequency fe) does not contain values above a
> given value w. Thus, f is smoothed is a sense:
> The idea I'm currently trying is to get uniform values between a and b
> at a rate w, then interpolate between the points to reach a sampling
> frequency fe. I'm not completely satisfied because the final curve
> reaches extreme values less frequently than central values (its
> distribution is a mode, I would like it to be flat). Have you got some
> ideas or isn't it clear ?
> Thanks in advance,
> Boris