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The Probability Density of X^2?

  1. Jun 17, 2015 #1
    Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out?

    Question:
    Let X be uniformly distributed random variable in the internal [ 0, 1]. Find the probability density of X^2?

    My trial:
    FY(y) = P(Y≤y) = P(X^2≤y) = P(X≤√y) = FX(√y) = ∫ (from 0 to √y) t dt = 0.5 y ⟹ 0.5 = fY(y).
    This is actually inspired by http://math.stackexchange.com/questions/...

    Solution on the textbook:
    y = x^2
    dy = 2x dx
    h(y)dy = 1 dx
    h(y) 2x dx = dx
    h(y) = 0.5/x = 0.5/√y
     
  2. jcsd
  3. Jun 17, 2015 #2

    haruspex

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    Why t dt, not just dt? And your long line of equalities is bracketed with FY(y)=...=fY(y), which is clearly not true. Have you missed a step in typing it out?
     
  4. Jun 17, 2015 #3
    Thanks get it

    Density function should be 1
     
  5. Jun 18, 2015 #4

    SammyS

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    I suppose that we will never know ,
     
  6. Jun 20, 2015 #5

    vela

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    They're not all equal signs; there's an arrow in there as well. The OP has ##F_Y(y) = y/2\ \Rightarrow\ f_Y(y) = 1/2##, which is, in fact, okay.
     
  7. Jun 20, 2015 #6

    haruspex

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    Ok. The ASCII character used for the arrow in the OP doesn't come out right on my iPad.
     
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