0

the term $x^n$ / n! equals the probability of the sum of n uniform random variables between 0 and 1 to be smaller or equal to x. so $x^(2n)$ / 2n! equals the probability of an even number of random variables between 0 and 1 to be smaller or equal to x. what does a succesive sum of these terms describe and why does it tend to cosh(x)? is there a probabilistic explanation/interpretation for this infinite sum? what probabilisitc problem could this infinite sum describing cosh(x) be solving? thanks alot!

Gilad
  • 61
  • It's very difficult to figure out what you are asking. In the power series for $\cosh,$ $x$ is an arbitrary real number. Since $x^{2n}/{2n}!$ can exceed $1,$ in what sense is it a probability?? The distribution of the sum of $n$ iid uniform$(0,1)$ variables definitely is not given by such a simple expression, either: see https://stats.stackexchange.com/questions/41467. – whuber Jul 27 '23 at 22:32
  • @whuber it could be in a combinatorical sense. PS. a similar question i'm also interested in is the probabilistic interpretation of the infinite sum of cos(x) which doesn't exceed 1, so if you have any ideas that would also help. – Gilad Jul 28 '23 at 00:25
  • Yes, there are combinatorial interpretations of many power series, especially the exponential, and if you force it you could probably come up with some probabilistic relationships, but this seems like it's inverting the usual question and answer process, as well as making your post more vague rather than clarifying it. To be on topic here, a post needs to formulate a definite question that plausibly has one correct objective answer. – whuber Jul 28 '23 at 13:39

0 Answers0