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The expression $x^n / n!$ appears in the infinite sum defining $e^x$ and similar terms in the sums defining $\cos(x)$, $\sin(x)$, etc. I would like to know if there is some combinatorial/probabilistic meaning or analogy to the term $x^n / n!$ and appropriately an example of a scenario of selection or decision making that could be represented by a consecutive sum of these terms similar to the infinite sums defining $e^x$ or $\cos(x)/\sin(x)$.

In other words: some probabalistic/combinatorical scenario whose calculation would converge to one of these functions ($e^x/\cos(x)/\sin(x)$).

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  • $\begingroup$ There are extremely many. One of the best known is a derangement. $\endgroup$
    – whuber
    Jul 5, 2023 at 14:31
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    $\begingroup$ @whuber I do not understand what $x^n/n!$ has to do with derangements, can you clarify? $\endgroup$ Jul 5, 2023 at 15:54

2 Answers 2

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If you divide every term by $e^{x}$ so the sum is $1$

then you get the Poisson distribution probability $e^{-x} \dfrac{x^n}{n!}$

of seeing $n$ items when the expected number of items is $x$.

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  • $\begingroup$ this @Henry still has the term e^−x . could you give some intuitive example of a set of calculations that converge to e^x? $\endgroup$
    – Gilad
    Jul 8, 2023 at 21:48
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If $X_1,X_2,\ldots,X_n$ are i.i.d $\text{Uniform}(0,1)$ random variables, then (check this for details)

$$P\left(\sum_{i=1}^n X_i \le x\right)=\frac{x^n}{n!}\quad,\, x\in [0,1]$$

In other words, $x^n/n!$ is the volume of the region $$\left\{(x_1,\ldots,x_n)\in \mathbb R^n: x_1,\ldots,x_n\ge 0, \sum_{i=1}^n x_i \le x\right\}$$ whenever $x\in [0,1]$. When $x=1$, this is related to the volume of a simplex.

We can also define a stopping rule associated with the $X_i$'s as

$$N=\min\left\{n: \sum_{i=1}^n X_i>x\right\} \quad,\, x\in [0,1]$$

Then, using the result above,

$$E(N)=\sum_{n=0}^\infty P(N>n)=\sum_{n=0}^\infty P\left(\sum_{i=1}^n X_i \le x\right)=e^x$$

This is in fact a popular exercise discussed here previously.

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  • $\begingroup$ What I like about this is the result is simple and yet the interpretation cannot be more manifesting. +1. $\endgroup$ Jul 6, 2023 at 6:22
  • $\begingroup$ @StubbornAtom what is the meaning of E(N)? $\endgroup$
    – Gilad
    Jul 8, 2023 at 21:46
  • $\begingroup$ @Gilad It is the expected number of $X_i$s when their cumulative sum first exceeds the given $x$ $\endgroup$
    – Henry
    Apr 2 at 10:23

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