I am trying to calculate the density of $\frac{Z_1^{\frac{2}{3}}}{Z_2}$,where $Z_1$ and $Z_2$ are independent normal random variables. It boils down to the calculation of this integral.Is there any way to calculate $\int_{0}^{\infty} \sqrt{y}x^{\frac{3}{2}}e^{-\frac{x^3y^3+x^2}{2}} dx$? Also,is there any other way to calculate density of,say, $Z_1^2/Z_2^3$
Calculating $\int_{0}^{\infty} \sqrt{y}x^{\frac{3}{2}}e^{-\frac{x^3y^3+x^2}{2}} dx$?
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$\begingroup$
integration
definite-integrals
normal-distribution
bivariate-distributions
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2as long as $y>0$ this integral is hopelessly divergent – 2017-01-30
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0I am extremely sorry,my post had quite a few errors. I have edited it now,the power of e is negative. – 2017-01-30
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2http://www.wolframalpha.com/input/?i=integrate%5Bx%5E(3%2F2)Exp%5B-x%5E3-x%5E2%5D,%7Bx,0,inf%7D%5D OMG!!! – 2017-01-30
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0What is the meaning of $Z_1^{2/3}$ if $Z_1<0$? – 2017-01-30
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0same as $ (-Z_1)^{\frac{2}{3}}$,I think – 2017-01-31