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Math Help - Distribution (FDD) of random process Xt

  1. #1
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    Distribution (FDD) of random process Xt

    Hello everyone!

    The question I have is:

    \xi and \eta are indpt standard normal random variables, we have a random process X_t ,where:

    X_t=(\xi-\eta)\sqrt{t},         t\geq0

    I would like to find out the distribution (one dimension FDD) of X_t ,for a fixed t>0

    Here is what I did:

    F_t(x)=P((\xi-\eta)\sqrt{t}\leq x)

    =\int P(\xi\leq\frac{x}{\sqrt{t}}+s|\eta=s)dF_\eta(s) by Total Probability Formula

    =\int P(\xi\leq\frac{x}{\sqrt{t}}+s)dF_\eta(s) by independence

    =\int F_\xi(\frac{x}{\sqrt{t}}+s)\cdot\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2} s^2} ds

    here is where I stuck...how should I deal with F_\xi(\frac{x}{\sqrt{t}}+s), or is there other ways to this kind of things?

    What about n dimensional FDD?

    Could anyone help me? :)
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  2. #2
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    Re: Distribution (FDD) of random process Xt

    I found a similar Q with answer, just post it on if it helps...

    \xi and \eta are indpt standard normal random variables

    X_t=(\xi+\eta)t, t\geq0

    Find the n dimensional FDD.

    and the answer looks like:

    if 0=t_1<t_2<...<t_k,

    F_{t_1,...,t_k}(x_1,...,x_k) =  \left\{\begin{matrix}0,&\mbox{ if }x_1<0,\\\Phi(\frac{1}{\sqrt{2}}min\{\frac{x_2}{t_  2},...,\frac{x_k}{t_k}\}),&\mbox{ if }x_1\geq 0,\end{matrix}\right

    if 0<t_1<t_2<...<t_k,

    F_{t_1,...,t_k}(x_1,...,x_k) = \Phi(\frac{1}{\sqrt{2}}min\{\frac{x_2}{t_2},..., \frac{x_k}{t_k}\})

    I understand where the min comes from, take 2D case, like above:

    P(\xi\leq\frac{x_1}{t_1}-\eta\wedge\frac{x_2}{t_2}-\eta)

    again using TPF:

    \int P(\xi\leq\frac{x_1}{t_1}-s \wedge\frac{x_2}{t_2}-s|\eta = s)dF_\eta(s)

    but how does this becomes
    F_{t_1,t_2}(x_1,x_2) = \Phi(\frac{1}{\sqrt{2}}min\{\frac{x_1}{t_1},\frac{  x_2}{t_2}\})???

    I believe the previous should have the similar FDD, so could anyone explain this to me??

    Thank you for your help
    Last edited by sweetadam; March 24th 2012 at 06:02 AM.
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  3. #3
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    Re: Distribution (FDD) of random process Xt

    Looks like I am answering my own question again...

    After a bit study, I found it is a very simple problem...maybe people won't type so much for a easy Q like this :P

    \xi and \eta are indpt standard normal random variables,

    therefore (\xi - \eta) \sim N(1, (\sqrt{2})^2), so that X_t \sim N(1, (\sqrt{2})^2)

    the 1D FDD is

    F_t(x)=P((\xi-\eta)\leq \frac{x}{\sqrt{t}})

    =P(\frac{(\xi-\eta)}{\sqrt{2}}\leq \frac{1}{\sqrt{2}}\frac{x}{\sqrt{t}})

    =P(z\leq \frac{1}{\sqrt{2}}\frac{x}{\sqrt{t}})

    = \Phi(\frac{1}{\sqrt{2}}\frac{x}{\sqrt{t}}})

    this could be easily extend to nD FDD
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