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Thread: Finding pdf question

  1. #1
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    Finding pdf question

    I have this random variable X that has a uniform distribution on [-pi, pi]. I want to find the pdf for Y = cosX.

    Here is my work so far:

    Fy(y)= P( Y < y ) = P ( cos X < y) =*P ( X < cos-1(y) ) + **P ( X > cos-1(y) ) (note: I got * since cos(x) is increasing on [-pi , 0] and ** since cos(x) is decreasing on [0 , pi]).


    =\displaystyle{\int_{-pi}^{cos^{-1}(y) } {\left( {1/2pi} \right)}  dx +  \int_{cos^{-1}(y)}^{pi} {\left( {1/2pi} \right)} dx}

    I am sure that I can't write this as one integral as the two cos-1(y) are not the same (and if I did it looks like I'll get 1)

    Since cos(x) is even I know that somehow these two integrals are equal but I just can't show it.

    Can someone please give me a hint?
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  2. #2
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    Re: Finding pdf question

    Hey JaguarXJS.

    You will have to restrict the domain so that the inverse function exists - and note that the inverse function should only go from 0 to +pi for inverse cosine.

    Can you take that into account for your answer?
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  3. #3
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    Re: Finding pdf question

    Quote Originally Posted by chiro View Post
    Hey JaguarXJS.

    You will have to restrict the domain so that the inverse function exists - and note that the inverse function should only go from 0 to +pi for inverse cosine.

    Can you take that into account for your answer?
    chiro, thank you for your reply. It's late here in New York and I'm turning in now. I will look at your response closely tomorrow.
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  4. #4
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    Re: Finding pdf question

    Quote Originally Posted by chiro View Post
    Hey JaguarXJS.

    You will have to restrict the domain so that the inverse function exists - and note that the inverse function should only go from 0 to +pi for inverse cosine.

    Can you take that into account for your answer?
    In the 1st integral I can let u=-x Then since -pi < x < 0 we get 0 < -x = u < pi as you you suggested. Then I will have
    F <sub> y </sub> (y) =\displaystyle{\int_{-cos^{-1}(y)}^{pi } {\left( {1/2pi} \right)}  du +  \int_{cos^{-1}(y)}^{pi} {\left( {1/2pi} \right)} dx}.

    Taking the derivative of both sides gives me

    f <sub> y </sub> (y)= [-(d/dy(-arccos(y))-(d/dy(arccos(y))]/(2pi) which I think should be 0. Where did I go wrong?
    Last edited by JaguarXJS; Dec 16th 2016 at 08:41 AM.
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