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for a simulation I'm using the continuous distribution $$F(x)=1-(1+x)e^{-cx} $$ for $x\geq 0$ with $c\geq 1$.

Do you know if this distribution has a name?

kjetil b halvorsen
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Jo R
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    From its very form you can see it is a mixture of a Gamma(1,c) and Gamma(2,c) distribution. For that reason alone I doubt it has a special name. – whuber Nov 28 '21 at 14:34

1 Answers1

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The corresponding density function can be found to be $$ f(x)= (cx+ c-1) e^{-cx}, \quad x\ge 0 $$ which can be written as a mixture $$ cx e^{-cx} + (c-1) e^{-cx} $$ which is a mixture of two gamma distributions, assuming $c\ge 1$ (necessary to avoid negative mixture weights). But from the form of the cdf (cumulative distribution function) given in the post, we can see that $c>0$ is enough, so that, interestingly, the mixture representation is not possible for all $c$.

There is probably no specific name for this distribution, at least, I did some searching and could not find any.

kjetil b halvorsen
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    The mixture coefficients are $1/c$ and $1-1/c$ for the Gamma$(2,c)$ and Gamma$(1,c)$ components, respectively. The reason to avoid $c\lt 1$ is not because negative weights are a problem--they can be perfectly valid--but because in that case, $f$ has negative values on the interval $[0, 1/c-1).$ – whuber Nov 29 '21 at 16:22