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Expectation of a Joint Distribution

  1. Jul 21, 2013 #1
    The below gives all the information I was given. I'm pretty sure my answer is right, but a part of me isn't, and that's why I'm asking here.

    1. The problem statement, all variables and given/known data
    Let (X,Y) have the joint pdf:

    [itex]f_{XY}(x,y)=e^{-y}, 0 < x < y < \infty[/itex]

    Find E(XY).

    2. Relevant equations

    [itex]E(XY)=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}xyf_{XY}(x,y)dxdy[/itex]


    3. The attempt at a solution
    Using the limits 0 < x < y and 0 < y < ∞,

    [itex]E(XY)=\int^{\infty}_{0}\int^{y}_{0}xye^{-y}dxdy=\frac{1}{2}\int^{\infty}_{0}y^{3}e^{-y}dy=3[/itex]

    Also, are my bounds correct?
     
  2. jcsd
  3. Jul 21, 2013 #2

    mfb

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    Looks good, and I can confirm the result of the integrations.
     
  4. Jul 21, 2013 #3
    Thank you very much. Now if I was trying to find [itex]f_{X}(x)[/itex] and [itex]f_{Y}(y)[/itex], which bounds would I use? I tried this (same info as above):

    Since [itex]f_{X}(x)=\int^{\infty}_{-\infty}f_{XY}(x,y)dy[/itex],

    [itex]f_{X}(x)=\int^{\infty}_{x}e^{-y}dy=e^{-x}[/itex]

    and

    [itex]f_{Y}(y)=\int^{y}_{0}e^{-y}dx=ye^{-y}[/itex]

    Assuming those are correct, which bounds do I use for E(X) and E(Y)? Because if I use the same bounds that I used for the marginal functions, I'll get the variable x in Y's mean, and the variable y in X's mean.
     
  5. Jul 21, 2013 #4

    mfb

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    Why? You can calculate E(X) and E(Y) with the original distribution and get a real number, or calculate it with your new distributions (in post 3) and get a real number.
     
  6. Jul 21, 2013 #5
    If I use 0 < x < y for E(X) and x < y < ∞ for E(Y), they'll each contain the other variable. For example,

    [itex]E(X)=\int^{y}_{0}xe^{-x}=1-e^{-y}(y+1)[/itex]

    I have a very strong feeling I'm using the wrong bounds...

    EDIT: If I use 0 < x < ∞ and 0 < y < ∞,

    [itex]E(X)=\int^{∞}_{0}xe^{-x}=1[/itex]

    [itex]E(Y)=\int^{∞}_{0}y^2e^{-y}=2[/itex]

    Is that correct?
     
    Last edited: Jul 21, 2013
  7. Jul 21, 2013 #6

    mfb

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    Your edit looks right.
     
  8. Jul 21, 2013 #7

    Ray Vickson

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    No. You seem to be plugging into formulas without understanding what you are doing. The marginal ##f_X(x)## has no ##y## in it, and the marginal ##f_Y(y)## has no ##x## in it. Here is how it works:
    [tex] f_X(x) \, \Delta x = P\{ x < X < x+\Delta x \} = \int_{y=x}^{\infty} f(x,y)\, \Delta x \, dy
    = \Delta x \times \int_{y=x}^{\infty} f(x,y)\, dy,[/tex]
    so ##y## is gone: it has been "integrated out".
     
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