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Math Help - Joint Density Questions

  1. #1
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    Joint Density Questions

    1) Let X and Y be two continuous random variables defined over the unit square. What does c equal if f x,y(x,y) = c(x^2 + y^2)?

    2) Suppose that X and Y have a bivariate uniform density over the unit square:
    f x,y(x,y) = c when 0 < x < 1, 0 < y < 1 and
    0, everywhere else
    a) Find c
    b) Find P (0 < X < 1/2, 0 < Y < 1/4)

    3) A point is chosen at random from the interior of a circle whose equation is x^2 + y^2 less than or equal to 4. Let the random variables X and Y denote the x and y coordinates of the sampled point. Find f x,y(x,y).

    4) For the following pdf, find fx(x) and fy(y)
    f X,Y(x,y) = 1/x, 0 < y < x < 1. (All these inequalities or less than or equal to, I just don't know how to do that on here.)
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  2. #2
    MHF Contributor matheagle's Avatar
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    (1) To find c, set...

    1=c\int_0^1\int_0^1(x^2 + y^2)dxdy

    (2a) since the base (unit square) has area 1, c=1, that way the volume is 1.
    (2b) just figure out the area of the rectangle (1/4)(1/2)=1/8.
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