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Thread: how to tell whether a function is dependent or independent problem

  1. #1
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    how to tell whether a function is dependent or independent problem

    how to tell whether a function is dependent or independent problem-uten-navn-ii.jpg

    In the green box in the attachment I have calculated the integral of the pdf of 24xy in the same manner as in the theory that leads up to 3.1 in the attachment. But I cant see any difference in that calculation that would explain that if 24xy is dependent why is not also the theory that leads up to 3.1 dependent. In the green box I would believe that $\frac{1}{2}(1-y)^2$ is $F_X(a-y)$ for a=1. I dont see any distinctions. Why is 24xy dependent while the theory up until 3.1 is independent?

    The theory is taken from Ross, A first course in probability.

    Tape image twice to read it.
    Attached Thumbnails Attached Thumbnails how to tell whether a function is dependent or independent problem-uten-navn-ii.jpg  
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  2. #2
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    Re: how to tell whether a function is dependent or independent problem

    Have you tried computing the marginal density functions from each joint density function?

    Idea: $\displaystyle X$ and $\displaystyle Y$ are independent IFF we can decompose the joint density function into the product of a function of $\displaystyle X$ only and a function of $\displaystyle Y$ only, for the entire range of $\displaystyle x$ and $\displaystyle y$, as given in the joint pdf.

    A definition of independence is that $\displaystyle X $ and $\displaystyle Y $ are independent continuous random variables if the product of the marginal pdf of $\displaystyle X $ and the marginal pdf of $\displaystyle Y$ equals the joint PDF of $\displaystyle X$ and $\displaystyle Y$ for all $\displaystyle x,y$. In mathematics, a definition is equivalent to an IFF (if and only if) statement! So we can use this idea to show whether or not $\displaystyle X$ and $\displaystyle Y$ are independent in our two situations.

    Situation 1:

    $\displaystyle f_{X,Y}(x,y)=24xy, 0<x<1, 0<y<1, x+y<1$

    $\displaystyle f_Y(x)=\int^{y=1-x}_{y=0} 24xy dy = 24x(\tfrac{y^2}{2})|^{y=1-x}_{y=0} = 12x(1-x)^2$

    $\displaystyle f_Y(y)=\int^{x=1-y}_{x=0} 24xy dx = 24y(\tfrac{x^2}{2})|^{x=1-y}_{x=0} = 12y(1-y)^2$

    $\displaystyle f_X(x) \cdot f_Y(y) \neq f_{X,Y}(x,y)$ because $\displaystyle 12x(1-x)^2\cdot 12y(1-y)^2 \neq 24xy$ for all $\displaystyle x,y$. Therefore, $\displaystyle X$ and $\displaystyle Y$ here are not independent.

    Situation 2:

    $\displaystyle f_{X,Y}(x,y)=6e^{-2x}e^{-3y}, 0<x<\infty, 0<x<\infty$

    $\displaystyle f_{X}(x)=6e^{-2x}\int^{y=\infty}_{y=0}e^{-3y} dy= 6e^{-2x}\left(\tfrac{e^{-3y}}{-3}\right)|^{y=\infty}_{y=0} = 6e^{-2x}\left(\tfrac{e^{-3y}}{3}\right)|^{y=0}_{y=\infty} = 2e^{-2x}$

    $\displaystyle f_{Y}(y)=6e^{-3y}\int^{x=\infty}_{x=0}e^{-2x} dx = 6e^{-3y}\left(\tfrac{e^{-2x}}{-2}\right)|^{x=\infty}_{x=0} = 6e^{-3y}\left(\tfrac{e^{-2x}}{2}\right)|^{x=0}_{x=\infty} = 3e^{-3y}$

    $\displaystyle f_X(x) \cdot f_Y(y) = f_{X,Y}(x,y)$ because $\displaystyle 2e^{-2x}\cdot 3e^{-3y} =6e^{-2x}e^{-3y}$. Therefore, $\displaystyle X$ and $\displaystyle Y$ here are independent.

    I hope this helps. Best of luck.
    -Andy
    Thanks from torgny
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  3. #3
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    Re: how to tell whether a function is dependent or independent problem

    Quote Originally Posted by abender View Post
    Have you tried computing the marginal density functions from each joint density function?

    Idea: $X$ and $Y$ are independent IFF we can decompose the joint density function into the product of a function of $X$ only and a function of $Y$ only, for the entire range of $x$ and $y$, as given in the joint pdf.

    A definition of independence is that $X $ and $Y $ are independent continuous random variables if the product of the marginal pdf of $X $ and the marginal pdf of $Y$ equals the joint PDF of $X$ and $Y$ for all $x,y$. In mathematics, a definition is equivalent to an IFF (if and only if) statement! So we can use this idea to show whether or not $X$ and $Y$ are independent in our two situations.

    Situation 1:

    $f_{X,Y}(x,y)=24xy, 0<x<1, 0<y<1, x+y<1$

    $f_Y(x)=\int^{y=1-x}_{y=0} 24xy dy = 24x(\tfrac{y^2}{2})|^{y=1-x}_{y=0} = 12x(1-x)^2$

    $f_Y(y)=\int^{x=1-y}_{x=0} 24xy dx = 24y(\tfrac{x^2}{2})|^{x=1-y}_{x=0} = 12y(1-y)^2$

    $f_X(x) \cdot f_Y(y) \neq f_{X,Y}(x,y)$ because $12x(1-x)^2\cdot 12y(1-y)^2 \neq 24xy$ for all $x,y$. Therefore, $X$ and $Y$ here are not independent.

    Situation 2:

    $f_{X,Y}(x,y)=6e^{-2x}e^{-3y}, 0<x<\infty, 0<x<\infty$

    $f_{X}(x)=6e^{-2x}\int^{y=\infty}_{y=0}e^{-3y} dy= 6e^{-2x}\left(\tfrac{e^{-3y}}{-3}\right)|^{y=\infty}_{y=0} = 6e^{-2x}\left(\tfrac{e^{-3y}}{3}\right)|^{y=0}_{y=\infty} = 2e^{-2x}$

    $f_{Y}(y)=6e^{-3y}\int^{x=\infty}_{x=0}e^{-2x} dx = 6e^{-3y}\left(\tfrac{e^{-2x}}{-2}\right)|^{x=\infty}_{x=0} = 6e^{-3y}\left(\tfrac{e^{-2x}}{2}\right)|^{x=0}_{x=\infty} = 3e^{-3y}$

    $f_X(x) \cdot f_Y(y) = f_{X,Y}(x,y)$ because $2e^{-2x}\cdot 3e^{-3y} =6e^{-2x}e^{-3y}$. Therefore, $X$ and $Y$ here are independent.

    I hope this helps. Best of luck.
    -Andy
    I just had to change from [tex] to $ in order to be able to read it. Thanks for the answer.
    Last edited by torgny; Dec 19th 2017 at 11:30 AM.
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  4. #4
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    Re: how to tell whether a function is dependent or independent problem

    Quote Originally Posted by torgny View Post
    I just had to change from [tex] to $ in order to be able to read it. Thanks for the answer.
    Hmm... Never had this issue before with LaTeX on this site -- thank you for making me aware. And I'm glad I could help!
    -Andy

    Edit: And it was rendering correctly after I posted. Perhaps I will link this to the LaTeX thread.

    Another edit: Explanation here: Need to buy a license to Codecogs?
    Last edited by abender; Dec 19th 2017 at 12:31 PM.
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