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Marginal Density of Coordinates Inside an Ellipse

  1. Feb 6, 2013 #1
    1. The problem statement, all variables and given/known data

    A point is chosen randomly in the interior of an ellipse:

    (x/a)^2 + (y/b)^2 = 1

    Find the marginal densities of the X and Y coordinates of the points.

    2. Relevant equations

    NA

    3. The attempt at a solution

    So this ought to be uniformly distributed, thus the density function for [itex](x,y)[/itex] is [itex]f_{x,y}[/itex] = [itex]1/∏ab[/itex] (where ∏ab is the area of the ellipse)

    So, to find the marginal density for x (and later for y), I realize that I just need to find the limits of integration and then go about my business. I believe that the limits of integration are
    [itex]-(b/a)\sqrt{a^2 - x^2}[/itex] and [itex]b/a\sqrt{a^2 - x^2}[/itex],

    since these should be the minimum and maximum values that y can take for any given x. Are these limits of integration and/or is my reasoning correct?

    As always, many thanks to all of you wonderful Homework Helpers!
     
    Last edited: Feb 6, 2013
  2. jcsd
  3. Feb 7, 2013 #2
    That seems entirely correct.
     
  4. Feb 7, 2013 #3
    Excellent, thanks!
     
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