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Math Help - Two questions

  1. #1
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    Two questions

    (1) show that -log(U) is exponential(1) if U is uniform(0,1)

    (2)Let X be exponential(2). Set Y=1-exp(-2Y) . Show Y is uniform(0,1)

    I thinks these two are quite similar. But I still dont know how to solve it. Anyone can? thx.
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  2. #2
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    Quote Originally Posted by solskjaer View Post
    (1) show that -log(U) is exponential(1) if U is uniform(0,1)

    [snip]
    Let Y = - \ln U and find the cdf of Y:

    F(y) = \Pr(Y \leq y)

    noting that 0 \leq u \leq 1 \Rightarrow 0 \leq y \leq +\infty

    = \Pr(- \ln U \leq y) = \Pr(\ln U \geq - y) = \Pr(U \geq e^{-y}) = \int_{e^{-y}}^1 1 \, du = 1 - e^{-y}.

    Then f(y) = \frac{dF}{dy} = e^{-y} for 0 \leq y \leq +\infty and zero elsewhere, as required.


    You should re-try (2).
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