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Let $S_n=\sum_{i=1}^n X_i$ be the sum of n iid random variables, and $M_N$ be $\max_{n\leq N} S_n$.

I know that given it is a simple random walk, i.e. $X_i=-1$ with probability $p$ and $-1$ with probability $1-p$, the probability of crossing a threshold $\mathbb{P}(M_N>c)$ can be calculated in closed form.

In general if we have a random variable $X_i$ with two outcomes not necessarily $-1$ and $1$, say $-1$ with probability $p$ and $8$ with probability $1-p$, is $\mathbb{P}(M_N>c)$ (equality, NOT inequality) still under closed form?

JT09
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  • Can you be more specific? You looking expected value or the new distribution? And a sum of any specific distributions? – cmitch Aug 28 '20 at 05:46
  • You define $M_n = \max_{n \leq N} S_n$ but there is likely a typo since $N$ appears nowhere, and the left-hand-side is a function of $n$, the right-hand-side I cannot interpret. I also do not understand your last sentence. – Michael Aug 28 '20 at 07:51
  • @Michael is it clear now? – JT09 Aug 28 '20 at 15:40
  • @cmitch I am looking for the probability of crossing a threshold. – JT09 Aug 28 '20 at 15:42

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