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Math Help - change of variable proof...

  1. #1
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    change of variable proof...

    Let X~Uniform[L,R]. Let Y=cX+d, where c<0. Prove that Y~Uniform[cL+d,cR+d]. (In particular if L=0 and R=1 and c=-1 and d=1, then X~Uniform[0,1] and also
    Y=1-X~Uniform[0,1])

    my attempt
    Y=cX+d
    X= (Y-d)/c since c<0
    (Y-d)/c ~ Uniform[L,R] since c<0
    isolating for Y gets you
    L <= (Y-d)/c <= R since c<0
    cL <= Y-d <= cR
    cL+d <= Y <= cR+d
    Y~Uniform[cL+d,cR+d]

    Y~Uniform[(-1)0+1,(-1)1+1]
    Y~Uniform[1, 0]
    1 <= -X+1 <= 0
    0 <= -X <= -1
    0 <= X <= 1
    X~Uniform[0,1]

    Y=1-X~Uniform[0,1]
    cX+d=1-X~Uniform[0,1]
    X~Uniform[0,1] = 1-(cX+d)
    X~Uniform[0,1] = 1-cX-d
    X~Uniform[0,1] = 1+X-1
    X~Uniform[0,1] = X

    is all this correct?



    also
    another question
    Let X~exponential(lambda). Let Y=cX where c>0. prove that Y~Exponential(lanbda/c)

    i don't understand how to do this proof.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Sneaky View Post
    Let X~Uniform[L,R]. Let Y=cX+d, where c<0. Prove that Y~Uniform[cL+d,cR+d]. (In particular if L=0 and R=1 and c=-1 and d=1, then X~Uniform[0,1] and also
    Y=1-X~Uniform[0,1])

    my attempt
    Y=cX+d
    X= (Y-d)/c since c<0
    (Y-d)/c ~ Uniform[L,R] since c<0
    isolating for Y gets you
    L <= (Y-d)/c <= R since c<0
    cL <= Y-d <= cR
    cL+d <= Y <= cR+d
    Y~Uniform[cL+d,cR+d]
    You can't do that!

    What you need to do is show that P(a<Y<b)=\frac{b-a}{cR-cL} when a,b \in  (cL+d, cR+d)

    (that the probability for any interval outside  (cL+d, cR+d) will be zero should follow)

    CB
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  3. #3
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    how do i do that?
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  4. #4
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    Quote Originally Posted by Sneaky View Post
    how do i do that?
    It is simple to do both questions using the same approach as I explained here: http://www.mathhelpforum.com/math-he...ge-159842.html

    Note that for the first question, note that the support of Y is cL+d \leq y \leq cR + d and that the cdf of Y is \displaystyle G(y) = R - \left( \frac{y - d}{c}\right) since c < 0 (all working is left for you to do).
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