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Math Help - Joint PDF

  1. #1
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    Joint PDF

    The joint probability distribution is given by,

    f_{(X,Y)}(x,y) = 2 if 0 \leq x \leq y \leq 1, 0 otherwise.

    (i) Determine the marginal density functions of X and Y. Check that it is indeed a pdf.

    (ii) Calculate the means \mathbb{E}X and VarX.

    (iii) Find the conditional density of X given Y = y for 0 \leq y \leq 1
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  2. #2
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    Quote Originally Posted by panda* View Post
    The joint probability distribution is given by,

    f_{(X,Y)}(x,y) = 2 if 0 \leq x \leq y \leq 1, 0 otherwise.

    (i) Determine the marginal density functions of X and Y. Check that it is indeed a pdf.

    (ii) Calculate the means \mathbb{E}X and VarX.

    (iii) Find the conditional density of X given Y = y for 0 \leq y \leq 1
    I will get you started:

    (i) f_X(x) = \int_{y=x}^{y=1} 2 \, dy and f_Y(y) = \int_{x=0}^{x=y} 2 \, dx.

    Show that \int_{0}^{1} f_X(x) \, dx = 1. Alternatively, show that \int_{0}^{1} f_Y(y) \, dx = 1.
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  3. #3
    MHF Contributor matheagle's Avatar
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    This uniform on half of the unit square has been posted here a few times.
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