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Math Help - maths stats - conditional density function

  1. #1
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    maths stats - conditional density function

    I'm just very confused about conditional density function. like the question is to show that Ey[E(X|Y)] = E(X)

    x=0 x=1 x=2 x=3
    y = 0 o.o4 o.o5 o.o6 o.o7
    y=1 o.o6 o.o8 o.o7 o.o9
    y=2 o.1 o.11 o.13 o.14

    well i think that E(X) is
    0x.04 + 0x.06 + 0x.1 + 1x.05 + 1x.08 + 1x.11 + 2x.06 + 2x.07 + 2x.13 + 3x.07 + 3x.09 + 3x.14 = 1.66

    and that E(X|Y) = P(x,y)/P(y)

    but the thing is that i dont know how to calculate P(x,y). Can someone explain it to me please?
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by continously confused View Post
    I'm just very confused about conditional density function. like the question is to show that Ey[E(X|Y)] = E(X)

    x=0 x=1 x=2 x=3
    y = 0 o.o4 o.o5 o.o6 o.o7
    y=1 o.o6 o.o8 o.o7 o.o9
    y=2 o.1 o.11 o.13 o.14

    well i think that E(X) is
    0x.04 + 0x.06 + 0x.1 + 1x.05 + 1x.08 + 1x.11 + 2x.06 + 2x.07 + 2x.13 + 3x.07 + 3x.09 + 3x.14 = 1.66

    and that E(X|Y) = P(x,y)/P(y) this is p(x|y) not E(X|Y)

    but the thing is that i dont know how to calculate P(x,y). Can someone explain it to me please?
    I don't understand your table.
    Is that p(x,y)?
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  3. #3
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    yeahh it is, but the thing is i really dont understand what my p(x,y) is supposed to be.. like is it just one number? like the row totals or something?

    Quote Originally Posted by matheagle View Post
    and that E(X|Y) = P(x,y)/P(y) this is p(x|y) not E(X|Y)

    ohhh and umm im really dumb, so what is the difference between p(x|y) and E(X|Y)? what is it meant to be? thanks
    Last edited by mr fantastic; March 25th 2009 at 12:11 AM. Reason: Merged posts
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  4. #4
    MHF Contributor matheagle's Avatar
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    YOUR table is hard to read, but it looks like all the numbers add to one.
    SO, that is p(x,y). For example P(X=0,Y=0)=.04....
    P(X=0|Y=0)=P(X=0,Y=0)/P(Y=0).

    Y can be 0,1,2 so there are 3 values for E(X|Y=y), depending on y

    E(X|Y=0)=\sum_{x=0}^3 xP(X=x|Y=0)
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  5. #5
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    ahhh icic thank you. :]. but how do i show that Ey[E(X|Y)] = E(X)?
    do i just do:

    E(X|Y=0)=\sum_{x=0}^3 xP(X=x|Y=0)
    E(X|Y=1)=\sum_{x=0}^3 xP(X=x|Y=1)
    E(X|Y=2)=\sum_{x=0}^3 xP(X=x|Y=2)

    and then add them up?
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  6. #6
    MHF Contributor matheagle's Avatar
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    E(X)=\sum_x\sum_y xp(x,y)

    YOU need to be careful with that y in

    E(E(X|Y))=\sum_y E(X|Y=y))P(Y=y)

    Just consider E(X|Y=y)) as g(y) and

    E(g(Y))=\sum_y g(y) P(Y=y)
    Last edited by matheagle; March 26th 2009 at 02:51 PM.
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  7. #7
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    ahhh okies, thank you!!
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