# Derivation for mixed distribution, Poisson-Lindley

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:

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

• 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 $y=\lambda(1+p).$ – whuber Aug 27 '19 at 12:27
• 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. – Tosh Aug 29 '19 at 3:12
• I'm sorry: that makes no sense. – whuber Aug 29 '19 at 12:37
• 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/… – kjetil b halvorsen Jan 12 '20 at 21:57
• Other relevant papers academia.edu/30523770/Lindley_distribution_and_its_application, jstor.org/stable/pdf/… – kjetil b halvorsen Jan 12 '20 at 23:50