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Thread: probability

  1. #1
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    probability

    Let $\displaystyle X $ be uniformly distributed over $\displaystyle (0,1) $. Let $\displaystyle Y = X^2 $. Find the cdf and pdf for the random variable $\displaystyle Y $, for $\displaystyle 0 \leq y \leq 1 $.

    So $\displaystyle f_{X}(x) = \begin{cases} 1 \ \ \ \ \ \ \ \ 0 < x < 1 \\ 0 \ \ \ \ \ \ \ \ \text{otherwise} \end{cases}$ and $\displaystyle F(x) = \begin{cases} 0 \ \ \ \ \ x < 0 \\ x \ \ \ \ \ 0 < x < 1 \\ 1 \ \ \ \ \ x > 1 \end{cases}$

    Then $\displaystyle X = \sqrt{Y} $ and so $\displaystyle f_{Y}(y) = \begin{cases} \frac{1}{2}y^{-1/2} \ \ 0 \leq \sqrt{y} \leq 1 \\ 0 \ \ \ \ \ \ \ \ \text{otherwise} \end{cases} $ and $\displaystyle F(y) = \begin{cases} 0 \ \ \ \ \ \sqrt{y} \leq 0 \\ \sqrt{y} \ \ \ \ \ 0 \leq \sqrt{y} \leq 1 \\ 1 \ \ \ \ \ \sqrt{y} \geq 1 \end{cases} $

    Then $\displaystyle P(X \leq 0.70) = F(0.70) = 0.70 $ and $\displaystyle P(Y \leq 0.70) = F(0.70) = 0.83 $

    Is this correct?
    Last edited by shilz222; Feb 19th 2008 at 09:42 AM.
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  2. #2
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    are the bounds correct?
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