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Probablity Question - Joint PDF Expectation/Variance

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data
    I have been given a joint PDF for X and Y, with ranges Y>X≥0.

    I need to find the E(x) and E(y).

    2. Relevant equations
    I know E(x) = ∫(x)*(f(x,y) dx and E(y) = ∫(y)*(f(x,y)) dy


    3. The attempt at a solution
    For ∫x*f(x,y) dx, i used the limits = x to ∞

    For ∫y*f(x,y) dy, i used the limits = 0 to y

    Is this correct?
     
  2. jcsd
  3. Nov 12, 2011 #2

    Ray Vickson

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    These are both wrong. EX = int x*f(x,y) dx dy, etc.

    RGV
     
  4. Nov 13, 2011 #3
    Im confused by what you meant above, but i meant to say;

    I have found f(x) by ∫f(x,y) dy, with the limits x to ∞.

    To then find the E(x) do i ∫x * f(x) dx, with the limits 0 to y?
     
  5. Nov 13, 2011 #4

    Ray Vickson

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    OK, this is equivalent to what I wrote. You really should use different symbols for the different distributions, such as g(x) = int f(x,y) dy and h(y) = int f(x,y) dx, or use f_X(x) instead of g(x) and f_Y(y) instead of h(y).

    As to your second question: the region in (x,y) space is {0 <= x <= y}, so yes, for any given x, y goes from x to infinity. However, once y has been "integrated out" it is no longer present, so NO, in EX = int x*f_X(x) dx, x does NOT go from 0 to y---there is no y now!. The variable x goes from 0 to infinity: when x was 5, y went from 5 to infinity, when x was 10 million, y went from 10 million to infinity, etc.

    RGV
     
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