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Math Help - Transformations of random variables with the uniform distribution

  1. #1
    Senior Member chella182's Avatar
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    Transformations of random variables with the uniform distribution

    This is probably hellishly easy, but I'm just not getting it.

    X is uniformly distributed over (0,1). For the random variable Y=-log_{e}(X)
    i) State the range of values Y can take.
    ii) Obtain the distribution function of Y, F_{Y}(y).
    iii) Obtain and identify the probability density function of Y, f_{Y}(y).

    Cheers for any help in advance.
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  2. #2
    MHF Contributor matheagle's Avatar
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    The range is (1,\infty)

    Let y exceed 1...

    F_Y(y)=P(Y\le y)=P(-\ln X\le y)=P(X\ge e^{-y})

     =\int_{e^{-y}}^1 dx=1-e^{-y} which is EXP(1)
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