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Math Help - Two independant and identically distributed uniform distributions

  1. #1
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    Two independant and identically distributed uniform distributions

    Let U and V be independent and identically distributed uniformly on the interval [0, 1].

    Show that for 0<x<1;

    P(x<V<U^2)=1/3 - x + (2/3)*x^3

    and hence write down the density of V conditional on the event that V<U^2.
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  2. #2
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    Quote Originally Posted by djones90 View Post
    Let U and V be independent and identically distributed uniformly on the interval [0, 1].

    Show that for 0<x<1;

    P(x<V<U^2)=1/3 - x + (2/3)*x^3

    and hence write down the density of V conditional on the event that V<U^2.
    1. Note that f(u, v) = 1 for 0 \leq u \leq 1, 0 \leq v \leq 1 and zero otherwise.

    2. Draw the square defined by u = 0, u = 1, v = 0 and v = 1 and shade the region corresponding to x < v < u^2.

    3. Then \Pr(x < V < U^2) = \int_{u = \sqrt{x}}^{u = 1} \int_{v = x}^{v = u^2} 1 \, dv \, du.

    I get \frac{1}{3} - x + \frac{2}{3} x^{\color{red}{3/2}}, NOT \frac{1}{3} - x + \frac{2}{3} x^{\color{red}{3}} ....
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  3. #3
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    my mistake.

    Thanks!
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