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Stochastic Processes

  1. Feb 16, 2008 #1
    1. The problem statement, all variables and given/known data
    I need someone to reassure me (or correct me) on this problem:

    The process [tex] X(t) = e^{At} [/tex] is a family of exponentials depending on the random variable A.
    Express the mean [tex] \eta(t) [/tex], the autocorrelation [tex] R(t_1,t_2) [/tex], and the first order density f(x,t) of X(t) in terms of the density [tex] f_a(a) of A [/tex]


    2. Relevant equations

    [tex] f(x,t) = \frac {\partial F(x,t)} {\partial x} [/tex]
    [tex] \eta (t) = \int_{-\infty}^{\infty} xf(x,t)dx [/tex]
    [tex] R(t_1,t_2) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x_1 x_2 [/tex]
    [tex]f(x_1,x_2;t_1,t_2)dx_1 dx_2 [/tex]


    3. The attempt at a solution

    [tex] f_a(a) = ae^a [/tex]
    [tex] f(x,t) = ae^{at} [/tex]
    [tex] \eta(t) = \int_{-\infty}^{\infty} a^2 e^{at} da [/tex]
    [tex] R(t_1,t_2) = \int_{-\infty}^{\infty} a^4 e^{at_1} e^{at_2} da
    [/tex]
     
    Last edited: Feb 16, 2008
  2. jcsd
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