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How to prove $\int_{0}^{\infty}\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp\left\{\frac{-\lambda(x-\mu)^2}{2\mu^2 x}\right\}dx=1$?

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Rather than asking for the answer, please delineate what concepts you are having trouble with in formulating your own proof. –  user8073 Dec 6 '12 at 20:24
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One method: look up the CDF (in Wikipedia, say) and differentiate it to prove it is correct. Then evaluate its rightmost limit. Both processes are purely mechanical. –  whuber Dec 6 '12 at 20:55
    
I know I can do this. There is actually a paper to find the cdf. I just want to find out how to do the integral directly because of my mathematical curiosity. –  ibictts Dec 6 '12 at 21:11
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2 Answers 2

You can use the closed form formula for the more general integral

$$\int_{ 0 }^{\infty} (ax^m)^{s} e^{\frac{-b(x-\mu)^2}{x}} = 2\,{a}^{s}{\mu}^{ms+1}{{\rm e}^{2b\mu}} {{\rm K_{m s+1}}\left(2\,b\mu\right)},$$

where $\rm{K}_{\nu}(x)$ is the modified Bessel function of the second kind.

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Remember that google is your friend. Here are some sources where you can find the proof

Youtube:

http://www.youtube.com/watch?v=g_bCDcNWcgU

Math StackExchange:

http://math.stackexchange.com/questions/248601/how-to-show-the-normal-density-integrates-to-1

Wikipedia:

http://en.wikipedia.org/wiki/Gaussian_integral

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These links seem to address a substantially different question of integrating a Gaussian density rather than the inverse Gaussian. –  whuber Dec 6 '12 at 20:52
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