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Math Help - Density Function

  1. #1
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    Density Function

    Consider the following joint probability density function of the random variable X and Y:

    f(x,y) = (3x - y) / 9 for 1 < x < 3 and 1 < y < 2

    a) Find the marginal distributions of X and Y.
    b) Are X and Y independent?
    C) Find P(X > 2)

    Not sure how to do this. I missed the class that went over it. Can someone show me how to do this? Thanks
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  2. #2
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    Quote Originally Posted by kenshinofkin View Post
    Consider the following joint probability density function of the random variable X and Y:

    f(x,y) = (3x - y) / 9 for 1 < x < 3 and 1 < y < 2

    a) Find the marginal distributions of X and Y.
    b) Are X and Y independent?
    C) Find P(X > 2)

    Not sure how to do this. I missed the class that went over it. Can someone show me how to do this? Thanks
    For a joint distribution of X and Y with density f(x,y) , the marginal distributions are:
     <br /> <br />
f_x(u)=\int_{sup(y)} f(u,y) dy <br />

    f_y(u)=\int_{sup(x)} f(x,u) dx

    where sup(y) is the set on which f(u,y) is defined (may be dependent on u) and similarly for sup(x)

    For X and Y to be independent we require that:

    f(u,v)=f_x(u)f_y(v)

    or slightly more confusingly:

    f(x,y)=f_x(x)f_y(y)

    RonL
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  3. #3
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    Well I was wondering if someone could show me how to do it. I have alot of them i can do in my book just wanted to see one done. I found what you typed in my book but am having a problem doing it.
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  4. #4
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    Quote Originally Posted by kenshinofkin View Post
    Consider the following joint probability density function of the random variable X and Y:

    f(x,y) = (3x - y) / 9 for 1 < x < 3 and 1 < y < 2

    a) Find the marginal distributions of X and Y.
    b) Are X and Y independent?
    C) Find P(X > 2)

    Not sure how to do this. I missed the class that went over it. Can someone show me how to do this? Thanks
    Quote Originally Posted by CaptainBlack View Post
    For a joint distribution of X and Y with density f(x,y) , the marginal distributions are:
     <br /> <br />
f_x(u)=\int_{sup(y)} f(u,y) dy <br />
    f_x(u)=\int_{sup(y)} f(u,y) dy =\int_{y=1}^2 (3u - y) / 9 dy=\left([3uy-y^2/2]_1^2\right)/9=\frac{u}{3}-\frac{1}{6}

    f_y(u)=\int_{sup(x)} f(x,u) dx
    f_y(u)=\int_{sup(x)} f(x,u) dx=\int_{x=1}^3 (3x - u) / 9 dx=\frac{4}{3}-\frac{2u}{9}


    RonL
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  5. #5
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    So for finding P(X > 2) what do I do. Plug 2 into something?
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  6. #6
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    Quote Originally Posted by kenshinofkin View Post
    So for finding P(X > 2) what do I do. Plug 2 into something?
    Since:

     <br />
P(X>2)=\int_{v=1}^2\int_{u=1}^2 f(u,v) du dv = \int_{u=1}^2 f_x(u) du <br />

    It is the integral of the marginal distrtribution of x from 1 to 2.

     <br />
P(X>2)=\int_{u=1}^2 f_x(u) du = \int_{u=1}^2 \left(\frac{u}{3}-\frac{1}{6}\right) du <br />

    RonL
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