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Probability (density function)

  1. May 11, 2009 #1
    A point is selected at random and uniformly from the region

    R = {(x,y): |x| + |y| <= 1 }

    Find the probability density function of the x-coordinate of the point selected at random.



    By definition f(x) = the integral of f(x,y) over all y values.

    after this I'm pretty much stuck. does f(x,y) = 1/4? I mean the definition is simple I think it's just me not knowing how to deal with the modulus signs.

    (Is the region a square(diamond) intersecting at (0,1)(1,0)(-1,0)(0,-1))

    Any help always appreciated.

    Thanks.
     
  2. jcsd
  3. May 11, 2009 #2
    Your diamond is right. You are right that [tex]f_{XY}(x,y)[/tex] is constant, but you have the wrong constant, because the area of that diamond is not 4.

    You could split the diamond into two halves, the case where x>0 and the case where x<0. Then you can obtain equations for the boundary lines without using absolute value. Your [tex]f_X(x)[/tex] could then be piecewise defined.

    When you are done, you could try to combine the piecewise defined [tex]f_X(x)[/tex] into one formula using absolute value, if you want, or simply leave it piecewise defined.
     
  4. May 11, 2009 #3
    f(x,y) = 1 (was looking at the diamond thinking it had side lengths of 2 :S, hope this is right now)

    as for the integrals im still having trouble...

    for x > 0

    f(x) = integral(from x-1 to -x+1) of 1 dy


    for x < 0

    f(x) = integral(from -x-1 to x+1) of 1 dy

    are these even remotely correct?

    I end up with 2-2x for x>0
    and 2+2x for x<0

    which if i combined them i get 2(1-|x|)

    the answer in the book is just 1-1|x|.... when i combine them do i drop the 2 :P
     
    Last edited: May 11, 2009
  5. May 11, 2009 #4
    Your f(x,y) is still wrong. What is the area of that diamond? When you fix it, you will get what they got.
     
  6. May 12, 2009 #5
    Thanks for the help Billy! Once i fixed up f(x,y) (dont ask me why it took me so long to realize that the area wasn't 1 :S) it all worked out!

    Thanks again!
     
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