for example the dynamical system is x_dot=f(x,t,theta), theta is a random variable, which can have Gaussian, Gamma, uniform, multi modal Gaussian or any other probability distribution.
is it possible to guess the type of pdf of x(t)? and by type, I mean just determine which type of pdf is it, not the characteristic or moments of pdf. for example if my theta~U[a,b] i need to know if x(t) is uniform or Gaussian for each t.
In general it is not possible, but you have to derive it for each function.
If solution is of the form x(t) = sin(w*t + theta) and theta follows uniform distribution (U[a,b]) then x is Gaussian.
For a linear system, if input is Gaussian output is also Gaussian.
Dear Dr Dharankar
What do you mean by "you have to derive it for each function"?
how to drive it? using sampling methods like Monte Carlo simulations? is there any analytical approach?
Hi Ali,
You can calculate the statistical moments of x from the moments of theta and from them you can reconstruct the corresponding probability distribution of x. See:
Working Paper Notes on the method of Cumulants for solution of linear equa...
Hope it helps,
Augusto
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