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Is there a well-known prob. distribution (or a combination thereof) that has pdf: $$\frac{1}{\sqrt{\pi}}t^{-1/2} e^{-t}$$ on $t \ge 0$ and $0$ everywhere else.

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up vote 6 down vote accepted

The pdf you listed is known as $\chi^2$-distribution with $1$ degree of freedom, i.e. it is the pdf $f_X(t)$ of $X=\frac12Z^2$, where $Z$ is the standard normal random variable.

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@did Thanks for fixing the typo. – Sasha Aug 27 '12 at 19:45
You are welcome. – Did Aug 27 '12 at 20:12
I get $\frac{t^{-1/2}e^{-t/2}}{\sqrt{2}\sqrt{\pi}}$. – Haderlump Aug 27 '12 at 20:37
@Haderlump The $\chi^2$ random variable is defined as $Z^2$, your density is that of $\frac{1}{2} Z^2$. See the edit. – Sasha Aug 27 '12 at 20:39

It's the $\Gamma(\frac{1}{2},1)$ distribution.

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