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Give a method for simulating from

$$F(x) =\left\{\matrix{ \frac{(1-e^{-2x}+2x)}{3},& 0 < x < 1\cr \frac{3-e^{-2x}}{3}, & 1 < x < ∞}\right.$$

(Work out the pdf, and try to write it as a mixture, with one of the components being an $Exp(λ)$ pdf )

Can someone help me here? What should I do after I differentiate the respective $F(x)$ and get $f(x)$.

Omran Kouba
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user9999
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  • Is this homework? What did you try? – kjetil b halvorsen May 06 '14 at 09:20
  • A sample question for my finals. I combined the two pdfs obtained from F'(x) and didn't know how to continue after that. – user9999 May 06 '14 at 11:17
  • How come we do not see what you did to follow the indication to "Work out the pdf, and try to write it as a mixture, with one of the components being an Exp(λ) pdf"? – Did May 19 '14 at 08:54

1 Answers1

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First differentiate $F$ to get the density function $f(x)$, then observe that the density can be writtes as $$ f(x) = \frac23 e^{-2x} + \frac23I(0<x<1). $$ That is a mixture between an exponential distribution (with mean $1/2$) and a uniform distribution on the unit interval, and can be simulated as such.

kjetil b halvorsen
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