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I’m looking for a density function $\rho(x)$ that is zero from $-\infty$ to, say, $a$, and then “bell-shaped” from $a$ to $\infty$. (In other words, it starts at zero at $a$, then rises until it reaches a peak, and then decreases again, and approaches zero as $x\to\infty$.)

Is there any famous and well-known density function like that, or do I have to construct my own using some piecewise trickery? (The latter is easy to do. However, it would be best if I could find a ready-made density function in the stockpile of well-known density functions.)

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How about $f(x) = (x-a)\exp(-(x-a))\mathbf 1_{(a,\infty)}$ or, more generally, $(1/n!)(x-a)^{n}\exp(-(x-a))\mathbf 1_{(a,\infty)}$, the $n$-th order Gamma density displaced $a$ to the right? – Dilip Sarwate Feb 8 '12 at 0:16
That would work (so would the lognormal distribution). However, I’m more looking for a well-known distribution that would have the number $a$ as one of its advertised parameters. Does such a distribution exist? – Topology Feb 8 '12 at 0:24
up vote 1 down vote accepted

Look some of the distributions listed by Wikipedia as supported on semi-infinite intervals. One example is the log-normal distribution but there are many more.

You may want to deal with $f(x-a)$ since many of these as described as supported on $[0,\infty)$. One which can incorporate the minimum as a parameter is Fréchet distribution.

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