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Suppose I order integers $1,2,3,...,n$ in some random permutation, such that the integers are lined up like $a_1, a_2, .., a_n$. I choose some random index $j$ and look at the left of $a_j$. I am interested to know what is the expected number of steps I need to take to the left for me to encounter another integer greater than or equal to $a_j$? By default, if no value greater than or equal to $a_j$ exists to the left, then the number of steps taken is $j$.

My approach is the following:

For $j=1$, $E(j)=1$; For $j=2$, there is 1 left neighbor $a_1$ which could has equal chance of either being greater or being smaller than $a_2$. So, the steps that will be taken is 1 step with probability 1/2 and 2 steps with probability 1/2. So, $E(2) = 1*1/2 + 2*1/2 = 2$. For $j=3$, there are 2 left neighbours $a_1, a_2$. The probability of $a_2>a_3$ is 1/2 and the probability $P(a_1>a_3|a_2<a_3)=1/2*1/2=1/4$.Thus, $E(3) = 1*1/2 + 2*1/4 + 3*(1-1/2-1/4)$ Thus, in general $$E(j) = \sum_{i=1}^{j-1}\frac{i}{2^i}+j\times\left(1-\sum_{i=1}^{j-1}\frac{1}{2^i}\right).$$

Does this seem reasonable?

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  • $\begingroup$ For $j=1$, $E(j)=1$. For $j=2$, $E(j)=2$, and so on, because the expectation of a constant is the constant. $\endgroup$ Oct 25, 2021 at 21:51
  • $\begingroup$ @paperskilltrees I suspect $E(j)$ is supposed to be the expected number of steps rather that the expectation of $j$. So for $j=2$ you have $E(j)$ as $\frac32$ $\endgroup$
    – Henry
    Oct 25, 2021 at 22:29
  • $\begingroup$ If $j > 2$ then the probability you take exactly two steps is not $\frac1{4}$ but $\frac{1}{6}$. Consider $P(a_1>a_3>a_2) = \frac16$ while $P(a_1>a_3\mid a_2<a_3) =\frac13$ $\endgroup$
    – Henry
    Oct 26, 2021 at 0:18
  • $\begingroup$ Does that mean that for $j>2$, the probability that you take exactly $n$ steps is $1/(n+1)!$? $\endgroup$
    – titanium
    Oct 26, 2021 at 2:11
  • $\begingroup$ @Henry Any non-standard notation should be explicitly introduced. Sometimes writing a question properly might help answering it (and in any case will help the efforts of others). $\endgroup$ Oct 26, 2021 at 2:58

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