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1

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

1

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.

2

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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