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Determining the joint probability density function

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data
    A process is defined as:

    X(t) = Asin(ωt+[itex]\phi[/itex]])

    where A and [itex]\phi[/itex]are random variables and ω is deterministic. A is a positive random variable.

    Determine the joint probability density function, PDF, of X(t) and X'(t) in terms of the joint PDF of A and[itex]\phi[/itex].


    2. Relevant equations

    PA(a) = ∫a FA[itex]\phi[/itex](a,[itex]\varphi[/itex])da
    P[itex]\phi[/itex]([itex]\varphi[/itex]) = ∫[itex]\phi[/itex] FA[itex]\phi[/itex](a,[itex]\varphi[/itex])d[itex]\varphi[/itex]

    joint PDF = PA(a)P[itex]\phi[/itex]([itex]\varphi[/itex])

    joint PDF of X(t) and X'(t) ??


    3. The attempt at a solution

    I'm confused on how to get a joint PDF of functions X(t) and X'(t) out of a function of A and [itex]\phi[/itex].

    Any suggestions would be greatly appreciated. It was suggested to assume there is a FA[itex]\phi[/itex](a,[itex]\varphi[/itex]). But I'm still confused.
     
    Last edited: Apr 11, 2012
  2. jcsd
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