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change of variable technique/Jacobian

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  • change of variable technique/Jacobian

    When using the change of variable technique with two variables, is it of the form

    g(y1,y2) |J| or g(y1,y2) |1 / J |

    where J is the jacobian. Another words, do I take the absolute value of the jacobian, or the absolute value of the inverse of the jacobian.

    I have two different books that conflict on this. if anyone knows I would greatly appreciate it.

    thanks and live long and prosper,

    mr. spock

  • #2
    Jacobian

    If (U,V) = ( U (X,Y), V(X,Y)) and f is the PDF of (X,Y) while g is the PDF of (U,V)
    first find the inverse function X(U,V), Y(U,V) and the Jacobian J for that inverse's
    derivative and then

    g(u, v) = f ( x(u,v), y(u,v) ) * | J |

    The reason why you may see 1/J is that the Jacobian of the derivative of
    (U,V) = ( U (X,Y), V(X,Y)) is the reciprocal of the Jacobian of (X,Y) = (X(U,V), Y(U,V))

    Yours,
    Krzys'
    Want to know how to pass actuarial exams? Go to: smartURL.it/pass

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