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Homework Help: Transformation of the uniform distribution

  1. Nov 20, 2011 #1
    1. The problem statement, all variables and given/known data

    I am told that X is a random variable with uniform distribution over [0,1]
    I need to find the mean and variance of log(X)

    2. The attempt at a solution
    I assume I must find the pdf of log(X) so I did this as follows;

    Let Y=log(X)
    Then to find the cumulative distribution function I considered;
    P(Y≤ y) = P(log(X)≤ y) = P( X < ey)

    I know that X is uniform over [0,1] so this is equal to
    ∫1.dy between ey and 0, where ey≤ 1

    This gives me that the cumulative distribution function = ey
    Which tells me that the probability density function is also ey

    Now what I am not sure of is what must y lie between, is this for y between eb and ea ?
  2. jcsd
  3. Nov 20, 2011 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    For X between 0 and 1, what is the range of log(X)?

    Anyway, to get the mean and variance of Y = log(X), you don't need the distribution of Y, although getting it is certainly one way of doing the problem.

  4. Nov 20, 2011 #3
    I think my lecturer wants me to do it this way. So y must be between - infinity and 0?
    Is my pdf correct now?
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