How to prove $\int_{0}^{\infty}\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp\left\{\frac{-\lambda(x-\mu)^2}{2\mu^2 x}\right\}dx=1$?
How to show that the inverse Gaussian density integrates to 1?
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calculus
definite-integrals
normal-distribution
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2Rather than asking for the answer, please delineate what concepts you are having trouble with in formulating your own proof. – 2012-12-06
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1One method: look up the CDF (in [Wikipedia](http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution), say) and differentiate it to prove it is correct. Then evaluate its rightmost limit. Both processes are purely mechanical. – 2012-12-06
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0I know I can do this. There is actually a [paper](http://amstat.tandfonline.com/doi/abs/10.1080/01621459.1968.10480942) to find the cdf. I just want to find out how to do the integral directly because of my mathematical curiosity. – 2012-12-06