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Thread: finding cdf of standard uniform

  1. #1
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    finding cdf of standard uniform

    A question about unifrom rv.

    If $\displaystyle X \;and\; Y$ are independent $\displaystyle standard \;uniform$ random variables. and we want to find the cumulative distribution of $\displaystyle X-2Y$

    to find the cdf here, can I suppose that $\displaystyle W=X-2Y,0<w<1$ and then find $\displaystyle P(W \leq w)$

    so, I have $\displaystyle P(W \leq w) = P(X-2Y \leq W) = P(Y \geq \frac{W-X}{2}) = 1 - P(Y < \frac{W-X}{2}) $

    and integrate $\displaystyle 1 - \displaystyle \int _0^w\;\int_0^{\frac{w-x}{2}}\;dy\;dx$

    is this right or am I wrong?
    Last edited by chutiya; Nov 23rd 2010 at 03:32 PM.
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  2. #2
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    Quote Originally Posted by chutiya View Post
    A question about unifrom rv.


    If $\displaystyle X \;and\; Y$ are independent $\displaystyle standard \;uniform$ random variables. and we want to find the cumulative distribution of $\displaystyle X-2Y$

    to find the cdf here, can I suppose that $\displaystyle W=X-2Y,0<w<1$ and then find $\displaystyle P(W \leq w)$

    so, I have $\displaystyle P(W \leq w) = P(X-2Y \leq W) = P(Y \geq \frac{W-X}{2}) = 1 - P(Y < \frac{W-X}{2}) $

    and integrate $\displaystyle 1 - \displaystyle \int _0^w\;\int_0^{\frac{w-x}{2}}\;dy\;dx$


    is this right or am I wrong?
    Wrong.

    For starters it should be $\displaystyle y < \frac{x - w}{2}$. Note also that the support of W is $\displaystyle -2 \leq w \leq 1$ so you will probably have to consider separate cases, that is, the cdf will be a hybrid function.
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  3. #3
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    Thank you. I always struggle with drawing the support. Since $\displaystyle -2 \leq w \leq 1$, we have to calculate the cdf for $\displaystyle X-2Y=W,\;-2 \leq w < -1$ and$\displaystyle -1 \leq w < 0 $, $\displaystyle 0\leq w< 1$ and then integrate, right?
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  4. #4
    MHF Contributor harish21's Avatar
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    correct!
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