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Math Help - probability independent distribution

  1. #1
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    probability independent distribution

    Hi,

    Ive been given this question but i just can figure it out.

    Suppose X,Y are Norm(1,1) and are independently and identically distributed.

    Find the distribution of :

    U = (X-1)/(absolute value(Y-1))

    Any help would be really appreciated.
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  2. #2
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    Quote Originally Posted by Number Cruncher 20 View Post
    Hi,

    Ive been given this question but i just can figure it out.

    Suppose X,Y are Norm(1,1) and are independently and identically distributed.

    Find the distribution of :

    U = (X-1)/(absolute value(Y-1))

    Any help would be really appreciated.
    I'll get you started with a possible approach:

    1. You know the joint pdf of X and Y, f(x,y).

    2. Consider the cdf of U.

    3. U = \frac{X - 1}{|Y - 1|} = \frac{X-1}{Y-1} if Y > 1 and \frac{X-1}{1 - Y} if Y \leq 1 so there are two cases to consider.

    Case 1: Y > 1.

    Case 2: Y \leq 1.

    Therefore:

    \Pr(U < u) = \Pr\left( \frac{X - 1}{Y - 1} < u | Y > 1\right) \cdot \Pr(Y > 1) + \Pr\left( \frac{X - 1}{1 - Y} < u | Y \leq 1 \right) \cdot \Pr(Y \leq 1)


     = \Pr(X - 1 < u[Y - 1] | Y > 1) \cdot \Pr(Y > 1) + \Pr(X - 1 < u[1 - Y] | Y \leq 1) \cdot \Pr(Y \leq 1)


    = \Pr(X < u [Y - 1] + 1 | Y > 1) \cdot \Pr(Y > 1) + \Pr(X < u [1 - Y] + 1 | Y \leq 1) \cdot \Pr(Y \leq 1) .

    Now integrate f(x, y) over the required regions and calculate each term.

    4. The pdf of U is given by g(u) = \frac{dF}{du}.
    Last edited by mr fantastic; November 18th 2008 at 01:56 AM.
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