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Deriving Joint Distribution Function from Joint Density (check my working pls!)

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Given:

    f(x,y) = x + y, for 0<x<1 and 0<y<1
    f(x,y) = 0, otherwise

    Derive the joint distribution function of X and Y.

    2. Relevant equations

    N.A.

    3. The attempt at a solution

    Using the definition, I obtained part of the joint distribution F(x,y) = (1/2)(xy)(x+y) for 0<x<1, 0<y<1. Leaving out the working for this as it is pretty standard. I am not sure if my next step is correct and so is hoping you guys can help check my logic...

    1) F(x,y) has to be defined for all (x,y) in R^2. So I have to consider all possible points. Correct?

    2) When 0<x<1 and y > or = 1, F(x,y) = left limit of F(x,y) as y tend to 1 = F(x,1) = (1/2)(x)(x+1)? I am guessing that values of y above 1 do not affect the joint distribution F(x,y) so it takes the same value as its left limit at the "boundary" of possible y values?

    3) By symmetry, when x > or = 1 and 0<y<1, F(x,y) = (1/2)(y)(y+1).

    4) Finally, for x > or = 1 and y > or = 1, F(x,y) = 1. And F(x,y) = 0 for all other values.

    Phew. I am most concern about step 2). The rest were included for completeness. Thanks!!!
     
  2. jcsd
  3. Apr 26, 2010 #2
    Making it shorter because I think the length is the reason why there are no replies...

    f(x,y) = x + y, for 0<x<1 and 0<y<1
    f(x,y) = 0, otherwise

    f = joint density. we want the joint distribution function, F.

    I only require asistance in verifying my logic for the expression for F over { 0<x<1 and y > or = 1} = B. I think that over region B, F(x,y) = left limit of F(x,y) as y tend to 1. Is this correct?

    Thanks!
     
  4. Apr 27, 2010 #3

    lanedance

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    i think its because the question isn't very clear - i'm not exactly sure what you mean by joint distribution function?

    the way I read your description, you have 2 dependent random vairbales X & Y, whose joint distribution function is:
    [itex] f_{X,Y}(x,y) = x + y [/itex] , for 0<x<1 and 0<y<1
    [itex] f_{X,Y}(x,y) = 0 [/itex] , otherwise

    Do you mean joint cumulative distribution function? defined by:
    [itex] F_{X,Y}(x,y) = P(X\leq x, Y\leq y) [/itex]

    I haven't looked at those much, but if so I would start by settting up an integral... start by looking at the area defined by your constraints...
     
    Last edited: Apr 27, 2010
  5. Apr 27, 2010 #4

    lanedance

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    if this is the case, the bounding bahviour of F as x (y) respectively go to 1 will be the marginal cumulative distributions of Y (X)
     
  6. Apr 27, 2010 #5
    Oops, sorry! I forgot to adjust for notation differences. In my course, they call the joint cumulative distribution function the distribution function and etc. But YES, what you laid out was precisely the question I require assistance on.

    I think I am fine with 90% of the working required for this question. Just a small part that I am not sure about:

    What is F(x,y) over the area { 0 < x < 1 and y > or = 1 } ?

    The double integral of f(x,y) that gives F(x,y) over 0 < x < 1 and 0 < y < 1 is (1/2)(xy)(x+y). The book I am using says for the area I am asking about, all we do is to find F(x,1). So it is saying that (1/2)(x)(x+1) is the answer to my question in bold.

    But it doesn't explain why. I am guessing we are taking the limit of F(x,y) as y tend to 1 from the left? (i.e. " left limit of F(x,y) as y tend to 1")
     
  7. Apr 27, 2010 #6

    lanedance

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    remember F(x,y) = P(X<=x, Y<=y)

    so for y>=1 F(x, y) will just reduce to the marginal cumulative distribution for x as P(Y<=1) = 1

    if you set up the integral it should be more clear
     
  8. Apr 27, 2010 #7
    Oh...I see. Thanks a lot man, you saved my *** several times on questions related to probability! Setting up the integral...?

    f(x,y) is defined for {0<x<1,0<y<1}

    by definition,


    [tex] F(x,y) = \int_{-\infty}^y \int_{-\infty}^x \! f(u,v) \, du dv [/tex]

    [tex] F(x,y) = \int_{0}^y \int_{0}^x \! f(x,y) \, dx dy [/tex]


    Right? But the result only works for 0<x<1 and 0<y<1? OR have I specified the limits wrongly?

    But...I go the answer I was looking for : for the region {0<x<1,y>or=1}, F(x,y) = cumulative distribution of x. Thanks!


    EDIT: Corrected terrible mistake with the double integral.
     
    Last edited: Apr 27, 2010
  9. Apr 27, 2010 #8

    lanedance

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    no worries - but be careful, x and y have a depndency, so its actually the marginal distribution of x see http://en.wikipedia.org/wiki/Marginal_distribution

    your integral is not quiet right either, note
    [tex] \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \! f(x,y) dx dy
    = \int_{0}^{1} \int_{0}^{1} f(x,y) dx dy = P(X\leq 1, Y \leq 1) = 1 [/tex]

    so have another think about the reigion of itegration to evaluate [itex] F_{X,Y}(x,y)= P(X\leq x, Y \leq x) [/itex]
     
  10. Apr 27, 2010 #9
    OMG terrible terrible mistake there. Not sure what I was thinking at the time of posting!

    Suppose to be:


    [tex] F(x,y) = \int_{-\infty}^y \int_{-\infty}^x \! f(u,v) \, du dv [/tex]

    [tex] F(x,y) = \int_{0}^y \int_{0}^x \! f(u,v) \, du dv [/tex]


    But the 2nd expression for F(x,y) only holds for the region {0<x<1,0<y<1} right? Beyond that, its either 0 or the respective marginal distribution?
     
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