论文标题
扰动和固定点的条件概率
Conditional Probability of Derangements and Fixed Points
论文作者
论文摘要
$ s_n $中的随机排列是一种危险的概率，众所周知是$ \ displaystyle \ sum \ limits_ {j = 0}^n（-1）^j \ frac {1} {j！} $。在本文中，我们考虑了$（k+1）^{st} $点固定的条件概率，鉴于第一个$ k $点中没有固定点。我们证明，当$ n \ neq 3 $和$ k \ neq 1 $时，此概率是$ k $和$ n $的降低功能。此外，证明了这种条件概率由$ \ frac {1} {n} - \ frac {k} {n^2（n -1）} $近似。鉴于第一个$ k $点中的$（k+1）^{st} $点是固定的，也获得了类似的结果。
The probability that a random permutation in $S_n$ is a derangement is well known to be $\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}$. In this paper, we consider the conditional probability that the $(k+1)^{st}$ point is fixed, given there are no fixed points in the first $k$ points. We prove that when $n \neq 3$ and $k \neq 1$, this probability is a decreasing function of both $k$ and $n$. Furthermore, it is proved that this conditional probability is well approximated by $\frac{1}{n} - \frac{k}{n^2(n-1)}$. Similar results are also obtained about the more general conditional probability that the $(k+1)^{st}$ point is fixed, given that there are exactly $d$ fixed points in the first $k$ points.
