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Math Help - int exp(-1/2 x^2) dx

  1. #1
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    int exp(-1/2 x^2) dx

    hi there, I'm reading a book on probability, but having trouble with following some equations. It's like this;
    \displaystyle \int_0^\infty \! \text{exp}\left( - \displaystyle \frac{1}{2}y^2 \right)\, dy \cdot\displaystyle \int _0^\infty \! \text{exp}\left( - \displaystyle \frac{1}{2}z^2 \right) \. dz \\  =\int_0^\infty\!\!\int_0^\infty  \text{exp} \left\{- \frac{1}{2}(y^2+z^2) \right\}dydz

    I don't know how this rewriting was made; please help.
    Thank you in advance.
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  2. #2
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    e^a \cdot e^b = e^{a+b}
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    thanks ,but that's what i understand(I should have asked more clearly...)

    what confuses me is that
    \int f(y)dy \cdot \int f(z)dz = \int \int {f(y)f(z)}dydz.

    would you help a little more? Thank you very much.
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  4. #4
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    This is taken directly from Stewart's book

    \int_c^d \int_a^b g(x)h(y)dxdy =  \int_c^d \left[\int_a^b g(x)h(y)dx\right]dy

    In the inner integral, y is constant, so h(y) is constant and we can write

    \int_c^d \left[h(y)\left(\int_a^b g(x)dx\right)\right]dy = \int_a^b g(x)dx \int_c^d h(y)dy

    since \int_a^b g(x)dx is constant.
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  5. #5
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    Quote Originally Posted by joll View Post
    thanks ,but that's what i understand(I should have asked more clearly...)

    what confuses me is that
    \int f(y)dy \cdot \int f(z)dz = \int \int {f(y)f(z)}dydz.

    would you help a little more? Thank you very much.
    Fubini's theorem - Wikipedia, the free encyclopedia
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