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My probability theory is a little rusty so I'm having trouble finding a nice expression for $P(Z<0)$ where $Z=X-Y,\quad X\sim\mathcal{N}(0,1)$ is a RV with a standard normal distribution, and $Y\sim \Gamma(k,\theta)$ is a RV with a gamma distribution. I seem to keep getting integrals that I'm unable to work with.

Edit: $X$ and $Y$ are both independent.

I_AM_ERROR
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    Are $X$ and $Y$ independent? – Feng Aug 30 '19 at 01:19
  • Which parameterization of the gamma is this? I'm aware of at least three that tend to be used. Is it the shape-rate parameterization or some other? – Glen_b Aug 30 '19 at 01:36
  • $X$ and $Y$ are indeed independent. The parameterization of gamma is flexible but I've been using $k$ as shape and $\theta$ as scale like the wikipedia page. – I_AM_ERROR Aug 30 '19 at 01:48
  • You'd get something like $$ \mathbb P(X<Y) = \int_0^\infty \int_{-\infty}^\infty \frac1{\sqrt{2\pi}}e^{-\frac12 x^2}\cdot \frac1{\Gamma(k)\theta^k}y^{k-1}e^{-\frac y\theta}\ \mathsf dx\ \mathsf dy, $$ but I don't believe this integral has a nice closed form. – Math1000 Aug 30 '19 at 02:05
  • Yup, that's basically what I keep getting. If there isn't a closed form I'll probably need to try another idea. Thanks for the help! – I_AM_ERROR Aug 30 '19 at 02:37

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