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Math Help - poisson process

  1. #1
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    poisson process

    Let {N(t) : t >=0} be a Poisson process of rate 1 and let T_{1} < T_{2} < ... denote the times of the points. Derive the pdf of Y=T_{2}/T_{4}

    So I've been trying to work with the distribution function. I have:

    P(T_{2}/T_{4} > x) = P(T_{2} > xT_{4}) = P(N(xT_{4}) \in \{0,1\})

    I get stuck there though and can't find a similar example.
    Last edited by Beaky; December 13th 2010 at 10:05 AM. Reason: typo
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  2. #2
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    Quote Originally Posted by Beaky View Post
    Let {N(t) : t >=0} be a Poisson process of rate 1 and let T_{1} < T_{2} < ... denote the times of the points. Derive the pdf of Y=T_{2}/T_{1}

    So I've been trying to work with the distribution function. I have:

    P(T_{2}/T_{4} > x) = P(T_{2} > xT_{4}) = P(N(xT_{4}) \in \{0,1\})

    I get stuck there though and can't find a similar example.
    It may help you to prove that  T_n -T_{n-1} are i.i.d. exponentials (where T_0=0). Now Y=\frac{T_2-T_1}{T_1}+1 where both T_2-T_1 and T_1 are independent exponentials. Now use the law of total probability and condition one of them.
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  3. #3
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    Sorry, typo in original post. It's Y=T_{2}/T_{4}. I'll try to use your advice anyways though.
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  4. #4
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    I get to P(\frac{T_{2}}{T_{4}}\le x)=P(\frac{T_{4}-T_{2}}{T_{2}}\ge \frac{1}{x}-1)

    But I get stuck here. I can't figure out how to apply conditioning and the law of total probability since the variables are continuous.
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  5. #5
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    Quote Originally Posted by Beaky View Post
    I get to P(\frac{T_{2}}{T_{4}}\le x)=P(\frac{T_{4}-T_{2}}{T_{2}}\ge \frac{1}{x}-1)

    But I get stuck here. I can't figure out how to apply conditioning and the law of total probability since the variables are continuous.
    I would suggest getting T_2-T_1 as it is exponential. The law of total probability says (for example)
    <br />
P(X+Y \in A)=\int P(X+y \in A)P(Y \in dy).<br />

    So suppose that X=T_2-T_1, then X is exponential with parameter one and
    <br />
P(Y \leq y)=\int P(T_4\geq x/y)e^{-x} dx<br />
    now you should try to figure out the law of T_4.
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