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Math Help - bivariate

  1. #1
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    bivariate

    given that f(x,y)= exp(-x^2+y^2/2), find probability of X+Y<1 when (X,Y) has this density.

    my book said that dividing f(x,y) by a constant K will produce a bivariate pdf. hence the P(X+Y<1) would be a double integral over f(x,y) /K.

    why do we need to divide by K and how do we know when to divide by K?
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    given that f(x,y)= exp(-x^2+y^2/2), find probability of X+Y<1 when (X,Y) has this density.

    my book said that dividing f(x,y) by a constant K will produce a bivariate pdf. hence the P(X+Y<1) would be a double integral over f(x,y) /K.

    why do we need to divide by K and how do we know when to divide by K?
    Assuming that the support is -oo < x < +oo and -oo < y < +oo then f(x, y) fails one of the two criteria for being a pdf unless it is divided by a constant whose value you should attempt to determine. Calculatng P(X + Y < 1) will require setting up and appropriate double integral and solving it.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Assuming that the support is -oo < x < +oo and -oo < y < +oo then f(x, y) fails one of the two criteria for being a pdf unless it is divided by a constant whose value you should attempt to determine. Calculatng P(X + Y < 1) will require setting up and appropriate double integral and solving it.
    How do you know it fails one of the two criteria? I would suppose u mean the one where their total sum is 1? Buthow does dividing it by1 solve this?
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    normalization and change of variables

    Quote Originally Posted by alexandrabel90 View Post
    given that f(x,y)= exp(-x^2+y^2/2), find probability of X+Y<1 when (X,Y) has this density.

    my book said that dividing f(x,y) by a constant K will produce a bivariate pdf. hence the P(X+Y<1) would be a double integral over f(x,y) /K.

    why do we need to divide by K and how do we know when to divide by K?
    You need to divide by K, because the density is not normalized; i.e., the integral of f(x,y) for -oo<x<+oo and -oo<y<+oo is not equal to 1. Determine K so that the integral of f/K is equal to 1.

    Next I do not think that you should just integrate x and y; e.g., over y from -oo to 1-x, and then over x from -oo to +oo. The reason is that you will get an error function for the integral over y, which makes it unnecessary difficult to do the integral over x. Instead it seems better to change to new variables; w=x+y and z=x-y looks good.
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