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Marginal distribution, double integral clarification

  1. Apr 30, 2013 #1
    1. The problem statement, all variables and given/known data

    (X,Y) is uniformly distributed over the area

    T = {(x,y): 0 < x < 2, -x < 2y < 0}

    Find the marginal probability functions ie [itex]f_{x}(x)[/itex] and [itex]f_{y}(y)[/itex].

    3. The attempt at a solution

    The thing I'm having trouble with is that y depends on x. Am I supposed to rewrite the boundaries for each marginal function? It feels like I'm doing things a roundabout way!

    F(x,y) = [itex]\int\int dx dy[/itex]

    I.e. to find f(y):

    -x < 2y < 0 [itex]\Leftrightarrow[/itex] x > -2y > 0 [itex]\Rightarrow[/itex] -2y < x < 2

    Which means I can integrate with respect to x from -2y to 2, leaving me with f(y) = 2 + 2y

    And if I instead want to find f(x):

    -x < 2y < 0 [itex]\Leftrightarrow[/itex] -(1/2)x < y < 0

    Which means I integrate with respect to y from -(1/2)x to 0, leaving me with f(x) = (1/2)x.

    Again, it feels pretty roundabout and I wanted to make sure I wasn't missing anything.
     
  2. jcsd
  3. Apr 30, 2013 #2
    Also, if I wanted to find E(X) and E(Y) here after finding f(x) and f(y), what interval should I integrate x*f(x) over, given that they are bounded by each other?
     
  4. Apr 30, 2013 #3

    haruspex

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    Your method of extracting f(x) and f(y) looks fine, and gets the right answers.
    For E(X), just integrate xf(x) over the full range of x, etc.
     
  5. Apr 30, 2013 #4
    Thanks Haruspex!
     
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