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the pmf of Negative Binomial-Inverse gaussian is given by :

$Pr(X=x)=\displaystyle\binom{r+x-1}{x}\left[\sum_{j=0}^{x}(-1)^{j}\binom{x}{j}\exp\left\{\frac{\psi}{\mu}\left[1-\sqrt{1+\frac{2(r+j)\mu^{2}}{\psi}}\right]\right\}\right]$.

with $x=0,1,2,...$ and $r,\mu,\psi>0$

how to prove all probabilities must between 0 and 1 and The sum of the probabilities must add up to 1 (2 requirement for discrete distribution)?

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