# How to prove the pmf of Negative Binomial-Inverse Gussian >0 and it's sum =1

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)?