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Math Help - finding cdf of standard uniform

  1. #1
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    finding cdf of standard uniform

    A question about unifrom rv.

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

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

    so, I have 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 1 - \displaystyle \int _0^w\;\int_0^{\frac{w-x}{2}}\;dy\;dx

    is this right or am I wrong?
    Last edited by chutiya; November 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 X \;and\; Y are independent standard \;uniform random variables. and we want to find the cumulative distribution of X-2Y

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

    so, I have 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 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 y < \frac{x - w}{2}. Note also that the support of W is -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 -2 \leq w \leq 1, we have to calculate the cdf for X-2Y=W,\;-2 \leq w < -1 and -1 \leq w < 0 , 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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