expected number of shared 1s between two binary strings from a given set

2012-10-02T04:53:23Z

Let say, I have two binary strings with length N, chosen from a set where there are $2^N-K,(K \ge 0)$ independent strings. What would be the expected number of Ones at the same index from two randomly picked strings from the set?

For example, 0010 and 1010, the number of ones at the same index is 1. Can it be by somehow related to the expected hamming distance between two binary strings?

------my own guess, some one could please verify---------

**sorry for the mess, unintentionally, the new problem was posted with this part..

having 1 at the i-th position is an independent event. So, let P(c_i=1) is the probability of having a common 1 at i-th position. Then, the expected number of shared 1s will be $\sum_{0..N-1} P(c_i)$. From the $2^N-K$ set, for ith position, count the number of 1s (denote $N^1_i$), and the number of 0s (denote $N^0_i$). Then $P(c_i)=\frac{N^1_i C 2}{(N^1_i+N^0_i) C 2}$. (C is combinations) when $N^1_i>=2$, otherwise $P(c_i)=0$.

For an example of {11110,1111,01110}, it gives me 3.66666, which sounds correct.

Answer by Granger for expected number of shared 1s between two binary strings from a given set

2012-10-02T18:42:20Z

Assuming I've understood the question correctly, this depends entirely on the distribution of 1's amongst the missing $K$ strings.

In particular, let $M = \{m_1,\ldots,m_K\}$ be the missing strings, and for $i \in \{0,\ldots,N-1\}$ let $k_i = \#\{ m \in M \mid m[i] = 1\}$, where $m[i]$ is the $i$-th bit of $m$.

Then the probability that two randomly chosen bitstrings from the remaining $2^N - K$ both have their $i$-th bit equal to $1$ is $\binom{2^{N-1} - k_i}{2}/ \binom{2^N - K}{2}$, and since these events are independent, the expected number of indices which are both 1 is $$ \sum_{i=0}^{N-1} \binom{2^{N-1} - k_i}{2}/ \binom{2^N - K}{2}. $$

expected number of shared 1s between two binary strings from a given set - MathOverflow

expected number of shared 1s between two binary strings from a given set

Answer by Granger for expected number of shared 1s between two binary strings from a given set