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I want to derive the Poisson Lindley Distribution.

$$ f_x(x|\lambda) = \frac{\lambda^{x-1}}{(x-1)!}e^{-\lambda} $$ $$f_x(x|p) = \frac{p^2}{(p+1)}(\lambda+1)e^{-\lambda p} $$

The Distribution of x, $f_x(x) = \int ^\infty _0 f_x(x|\lambda) f_\lambda(\lambda) \ d\lambda$

Then $$ f_x(x) = \frac{p^2}{(p+1)(x-1)!} \int_0^\infty \lambda^{x-1}(\lambda+1) e^{-\lambda (1+p)} \; d\lambda $$

I broke it as sum of 2 integrals:

$$ f_x(x) = \frac{p^2}{(p+1)(x-1)!}\left[ \int^\infty_0 \lambda^xe^{-\lambda(1+p)} \; d\lambda \ +\int^\infty_0 \lambda^{(x-1)}e^{-\lambda (1+p)} \; d\lambda \right] $$

As I solve $\int ^\infty _0 \lambda^xe^{-\lambda(1+p)} \ d\lambda = \frac{\lambda^x e^{-\lambda(1+p)}}{-(1+p)} +\frac{1}{1+p} \int ^\infty _0 \lambda^{x-1}e^{-\lambda(1+p)} \ d\lambda $

Then rewriting $f_x(x):$ $$f_x(x) = \frac{p^2}{(p+1)(x-1)!} \int^\infty_0 \frac{p+2}{p+1} \lambda^{x-1} e^{-\lambda (1+p)} \; d\lambda $$

I am unable to proceed further to show that the Poisson-Lindley distribution is:

$$\frac{p^2}{(p+1)^{x+3}}(p+2+x)$$

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    $\begingroup$ You lost me at the outset with your two different, conflicting definitions of "$f_x$." It appears you meant to write "$f_\lambda(\lambda\mid p)$" in the second instance whereas in the integral you needed to write "$f_\lambda(\lambda\mid p)$" instead of "$f_\lambda(\lambda).$" As far as the mathematics goes, you're just evaluating the sum of Gamma integrals: you can look them up or use the substitution $y=\lambda(1+p).$ $\endgroup$ – whuber Aug 27 '19 at 12:27
  • $\begingroup$ Right, I was supposed to write $f_\lambda (\lambda |p)$, but $\lambda$ is independent of $p$, so can write it as $f_\lambda (\lambda)$. Thanks for the hint. Shall try it. $\endgroup$ – Tosh Aug 29 '19 at 3:12
  • $\begingroup$ I'm sorry: that makes no sense. $\endgroup$ – whuber Aug 29 '19 at 12:37
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    $\begingroup$ I tried to fix your $\LaTeX$ markup, please check. But there are still errors/inconsistencies, please fix. As for your question see s3.amazonaws.com/academia.edu.documents/50968086/… $\endgroup$ – kjetil b halvorsen Jan 12 at 21:57
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    $\begingroup$ Other relevant papers academia.edu/30523770/Lindley_distribution_and_its_application, jstor.org/stable/pdf/… $\endgroup$ – kjetil b halvorsen Jan 12 at 23:50

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