Matt a écrit : > On Feb 10, 3:21 pm, Boris Gourévitch <bo...@SUS-AU-SPAMpi314.net> > wrote:
> >> 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 > "derivable"... > yes, sorry, typical french mistake... ;-)
>> 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. >
yes, you're right, I was too quick, the probability should be proportional. I guess that another way to express it correctly is: the distribution function of f(t) is a uniform distribution on [a,b].
Maybe a more rigorous way to present the problem is: is there a stochastic process Xt, t in R+ (resp. N), differentiable (resp. with a low pass spectrum) and so that the probability density function of Xt is uniform ? (A Brownian process is continuous and follows a gaussian distribution if I remember)
Intuitively, I'm looking for a curve that would freely move with time between a and b and whose distribution on [a,b] would be uniform. Thanks in advance Boris