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I have the following distribution but I am unsure if it is a common distribution. Assume that $\alpha +\beta>0$. I have searched but nothing comes up. Any help would be appreciated!

\begin{equation} p(x) = \frac{\sqrt{\beta}}{\sqrt{\pi}}e^{-\frac{\alpha^2}{4 \beta}} e^{-\alpha x-\beta x^2} I_{-\infty,\infty}(x) \end{equation}

dsmalenb
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1 Answers1

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Yes, this is a Gaussian distribution: just complete the square in the exponent. After doing so, you will find that it is $N(\mu,\sigma)$ where $\mu=-\alpha/2\beta$ and $\sigma=1/\sqrt{2\beta}$ (assuming that $\beta>0$, otherwise your formula doesn't give a probability density).

pre-kidney
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