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Math Help - Probability Integral

  1. #1
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    Probability Integral

    In probability theory the important Gaussian Distribution often leads to the integral X=\int_{-\infty}^{\infty} e^{-x^2} dx .

    In case you ever wondered where it comes from, here (informal but nice*) is a non-complex analysis demonstration.

    We begin by considering the integral,
    Y=\iint_{\mathbb{R}^2} e^{-(x^2+y^2)} \ dA

    This is the integral over the entire plane \mathbb{R}^2.

    Now there are two ways to get the entire plane.

    Method 1: For a,b>0 we can create a rectangle -a\leq x\leq a \mbox{ and }-b\leq x\leq b. Now we make a\to \infty \mbox{ and }b\to \infty. That infinite rectangle will take the entire plane.
    In other words, Y=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)} \ dy \ dx.
    Now, the important realization is that this double integral has "seperated variables" meaning f(x,y)=g(x)h(y) and we can write \int_{-\infty}^{\infty} e^{-x^2} dx \cdot \int_{-\infty}^{\infty} e^{-y^2} dx. Hence, Y = X^2 \implies X = \sqrt{Y} (because it cannot be negative).

    Method 2: We notice the expression x^2+y^2 in the double integral and try a polar coordinates substitution. If we let r^2 = x^2+y^2, we can let 0\leq \theta \leq 2\pi and r\to \infty. In other words, we draw a circle at the orgin with infinite radius, and this will take on values on the entire \mathbb{R}^2 plane.
    Thus, we end up with after a change to polar coordinates,
    Y = \int_0^{\infty} \int_0^{2\pi} e^{-r^2} \ r d\theta \ dr = \lim_{r\to \infty} 2\pi \int_0^r re^{-r^2}dr = 2\pi \left( \frac{1}{2} \right) = \pi

    Thus, Y=\pi which implies X=\sqrt{\pi}.

    And hence,
    \int_{-\infty}^{\infty} e^{-x^2} \ dx = \sqrt{\pi}


    *)I am sure it can be made formal. But I do not know how do it because I am not familar with the theory of real multiple integration.
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  2. #2
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    In my calculus textbook, a squeeze technique is used to prove the result via double integrals in polar coordinate.
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