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Math Help - Density Question

  1. #1
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    Density Question

    1) Assume that (X,Y) has the following density function
    f(x,y) = 2e^-(x+y) when 0 < x < y
    a) Find the marginal density function fx(x) and fy(y)
    b) Find the probability that P (Y < 2X)
    c) Find P(Y < 1 given X < 1)
    d) Find P(Y < 1 given X = 1/2). Also the E(Y given X=1/2).
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  2. #2
    MHF Contributor matheagle's Avatar
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    a) f_X(x)=2e^{-x}\int_x^{\infty}e^{-y}dy and f_Y(y)=2e^{-y}\int_0^ye^{-x}dx

    b) Draw and integrate

    c) Use P(A|B)={P(AB)\over P(B)}

    d) Find f(y|x) and integrate.
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  3. #3
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    For b), what do I draw?

    For c), isnt it P(A union B)/ P(B)? And how do I know what something like P(B) is? Integrate from 0 to 1 with X? Not exactly sure what to ask, but I'm still confused.

    And for d), for the expectation part, do I just integrate with y * f(Y given x)?
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  4. #4
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    mr fantastic's Avatar
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    Quote Originally Posted by Janu42 View Post
    For b), what do I draw? Mr F says: Integrate f(x, y) over the region that is under the line y = 2x but above the line y = 0 (that is, the x-axis).

    For c), isnt it P(A union B)/ P(B)? Mr F says: No it's not. Did you look in your class notes at all. It's intersection (which is what matheagle's notation says) not union.

    And how do I know what something like P(B) is? Integrate from 0 to 1 with X? Mr F says: Yes. Use the marginal density function.

    Not exactly sure what to ask, but I'm still confused.

    And for d), for the expectation part, do I just integrate with y * f(Y given x)? Mr F says: What do your class notes say to do?
    ..
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    ..

    Sorry, yeah I meant intersection, but OK. I was just confused because it says AB and doesn't have the intersection symbol but I get it thanks.
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