We know that the density function in the form of $\varphi=e^{-x^2}$ stands for normal distribution. What is the name of the Distribution with density function $\varphi=e^{-|x|}$?
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5$\begingroup$ Double exponential or Laplace, en.wikipedia.org/wiki/Laplace_distribution if you multiply by $1/2$. $\endgroup$– JohnKJun 10, 2017 at 13:05
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2$\begingroup$ What is $\varphi$ intended to represent here? You seem to call it a density function but it can't be (since neither integrates to 1). Are you saying something more like $\varphi \propto f$ where $f$ is a density? Alternatively you could keep the term "density" but insert a $\propto$ in place of $=$. $\endgroup$– Glen_bJun 11, 2017 at 0:08
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1 Answer
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That's a particular case of the Laplace distribution, $f(x) = \frac{1}{2b}\exp{\left(-\frac{|x-\mu|}{b}\right)}$. You can find more information in most text books or in the Wikipedia.
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$\begingroup$ +1 Double exponential is a common alternative name (though it has a rarer second meaning as well). I don't know that I'd say "most text books"; many perhaps -- there are shelves and shelves of introductory statistics books that never mention the Laplace. If you look at books that have titles with "mathematical statistics" in them or probability texts you might spot it though. $\endgroup$– Glen_bJun 11, 2017 at 0:02
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$\begingroup$ Hummm, yes, I think you are right. Laplace distribution is not so ubiquitous in entry-level books, I gave perhaps the wrong impression saying that. Still there is the Wikipedia and, as you mention, books more mathematically oriented. Thank you for editing my answer. $\endgroup$ Jun 11, 2017 at 9:16