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Math Help - Probability Density Function Question

  1. #1
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    Probability Density Function Question

    If someone could review my work on this exercise

    When I use f X(x) is the pdf of X and fY(y) is the pdf of Y.

    Consider X a random variable with pdf

    f X(x) = (3/8) * (1-x)^2 if -1<=x<=1

    and 0 otherwise.

    Find the pdf of Y = 1-X^2

    My workings:

    If -1<=x<=1 then 0<=Y<=1

    Then P(Y<=y) = P(1-X^2<=y) = P(X^2>=1-y) = P(X>=sqrt(1-y))

    =1-P(X<=sqrt(1-y))

    =1 - (3/8) * (1 - sqrt(1-y)) ^ 2 for 0<=y<=1

    and 0 otherwise


    Now f Y (y) = d/dy (P(Y<=y))

    = (3/4) * (1-sqrt(1-y)) / (2*sqrt(1-y)) for 0<=y<=1 and 0 otherwise!



    Or should I say P(X^2>=1-y) = 1-P(X^2<=1-y) = 1- P(-sqrt(1-y) <=X<= +sqrt(1-y)) and then how do I proceed?
    Last edited by Darkprince; December 3rd 2011 at 09:07 AM.
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    Re: Probability Density Function Question

    Quote Originally Posted by Darkprince View Post
    If someone could review my work on this exercise

    When I use f X(x) is the pdf of X and fY(y) is the pdf of Y.

    Consider X a random variable with pdf

    f X(x) = (3/8) * (1-x)^2 if -1<=x<=1

    and 0 otherwise.

    Find the pdf of Y = 1-X^2

    My workings:

    If -1<=x<=1 then 0<=Y<=1

    Then P(Y<=y) = P(1-X^2<=y) = P(X^2>=1-y) = P(X>=sqrt(1-y))

    =1-P(X<=sqrt(1-y))

    =1 - (3/8) * (1 - sqrt(1-y)) ^ 2 for 0<=y<=1

    and 0 otherwise


    Now f Y (y) = d/dy (P(Y<=y))

    = (3/4) * (1-sqrt(1-y)) / (2*sqrt(1-y)) for 0<=y<=1 and 0 otherwise!



    Or should I say P(X^2>=1-y) = 1-P(X^2<=1-y) = 1- P(-sqrt(1-y) <=X<= +sqrt(1-y)) and then how do I proceed?
    cdf = G(y) = \Pr(Y < y) = \Pr(1 - X^2 < y)

     = \Pr(X > \sqrt{1 - y}) + \Pr(X < -\sqrt{1 - y}) = ....


    Therefore pdf = g(y) = \frac{dG}{dy} ..... for 0 \leq y \leq 1 and 0 otherwise.
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    Re: Probability Density Function Question

    Quote Originally Posted by mr fantastic View Post
    cdf = G(y) = \Pr(Y < y) = \Pr(1 - X^2 < y)

     = \Pr(X > \sqrt{1 - y}) + \Pr(X < -\sqrt{1 - y}) = ....


    Therefore pdf = g(y) = \frac{dG}{dy} ..... for 0 \leq y \leq 1 and 0 otherwise.
    So I should say that:

    P(X^2>=1-y) = 1-P(X^2<=1-y) = 1- P(-sqrt(1-y) <=X<= +sqrt(1-y)) = 1 - [P(X>= - sqrt(1-y)) + P(X<=sqrt(1-y))] = 1-[ 1- P(X<= - sqrt( 1 -y) + P (X<= sqrt (1-y))]

    =P(X<= -sqrt(1-y)) - P(X<= sqrt(1-y))
    Last edited by Darkprince; December 4th 2011 at 03:50 PM.
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    Re: Probability Density Function Question

    But to find P(X<= - sqrt(1-y)) and P(X<=sqrt(1-y)) I have to find the Cdf F X (x)

    The thing that worries me is that since -1<=x<=1 if I integrate f X (x) from -1 to 1 I will get a real number and not a cdf in terms of x. So what should I do?


    Then the cdf will be f Y (y) = d (F Y (y)) / dy and is a matter of calculation.
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    Re: Probability Density Function Question

    Quote Originally Posted by Darkprince View Post
    But to find P(X<= - sqrt(1-y)) and P(X<=sqrt(1-y)) I have to find the Cdf F X (x)

    The thing that worries me is that since -1<=x<=1 if I integrate f X (x) from -1 to 1 I will get a real number and not a cdf in terms of x. So what should I do?


    Then the cdf will be f Y (y) = d (F Y (y)) / dy and is a matter of calculation.
    The cdf is given by \int_{-1}^{-\sqrt{1-y}} \frac{3}{8} (1 - x)^2 dx + \int^{1}_{\sqrt{1-y}} \frac{3}{8} (1 - x)^2 dx.

    Your job is to do the integrations and then differentiate the result with respect to y.
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    Re: Probability Density Function Question

    Quote Originally Posted by mr fantastic View Post
    The cdf is given by \int_{-1}^{-\sqrt{1-y}} \frac{3}{8} (1 - x)^2 dx + \int^{1}_{\sqrt{1-y}} \frac{3}{8} (1 - x)^2 dx.

    Your job is to do the integrations and then differentiate the result with respect to y.

    If I say that I have to do 1- P(-sqrt(1-y) <=X<= +sqrt(1-y)) could I do (1 - the integral from -sqrt(1-y) to +sqrt(1-y) of the cdf)? i.e just one integral?
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    Re: Probability Density Function Question

    Quote Originally Posted by Darkprince View Post
    If I say that I have to do 1- P(-sqrt(1-y) <=X<= +sqrt(1-y)) could I do (1 - the integral from -sqrt(1-y) to +sqrt(1-y) of the cdf)? i.e just one integral?
    Yes.
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    Re: Probability Density Function Question

    Thank you
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