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$$\begin{split}\sum_{k=0}^{[nx]}\binom{n}{k}s^k(1-s)^{n-k}& =\sum_{k=0}^n \mathbb{1}_{k\leq [nx]} \binom{n}{k}s^k(1-s)^{n-k} \\ & = \sum_{k=0}^n \mathbb{1}_{k\leq nx} \binom{n}{k}s^k(1-s)^{n-k} \\ & = \mathbb{P}(\mathcal{B}(n,s)\leq nx)\\ & = \mathbb{P}\left(\frac{\mathcal{B}(n,x)}{n}\leq x\right)\end{split}$$ where $\mathcal{B}(n,s)$ is a ...


The answer is undefined as the probability distribution provided is invalid, the probabilities don't sum up to one, so there's not much to expand on here.


Assuming you've used this definition for the NIG distribution and that you've managed to come up with estimates $(\hat{\alpha}, \hat{\beta}, \hat{\mu}, \hat{\delta} )$ of the individual NIG parameters, your question boils down to: "How to simulate paths from the global log-return process $R_t = \ln(S_t/S_0)$ for all $t \in [0,T]$, assuming i.i.d. ...

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