In the kindergarten group, n children are of different sizes. They came into the circle at random. The child will say that he is tall if he is higher than his two neighbors. Find the mathematical expectation and variance of children who will call themselves high.
We need to split random variable into following sum and calculate: $$E(\xi) = E(\xi_1) + ...+E(\xi_n)$$ where $$\xi_i = 1$$ if child said he is high. So basically the reason why I am stuck here is that if |i - j| = 1 and the i-th kid said that he is high then $$E(\xi_j) = 0 $$ but if |i - j| >= 2 then it is okay and both Expectations equal to 1/4
so I am stuck with the following steps because of that and can not come up with a solution = (
please, help me out!
To elaborate on the discussion in the comments:
The quick answer is to look at the slots, numbered by $i\pmod n$. For each slot $i$ consider the kids in slots $i-1, i, i+1$. One of these is the shortest of the three and the probability that the shortest child is in slot $i$ is thus $\frac 13$. As there are $n$ slots, the answer is $\frac n3$.
Alternative argument: Rank the children by height. The probability that the $i^{th}$ tallest is taller than their neighbors is $\binom {i-1}2\big /\binom {n-1}2$. Thus the expected number of "tall" children is $$\frac 1{\binom {n-1}2}\times \sum_{i=1}^{n}\binom {i-1}2=\frac 1{\binom {n-1}2}\times \binom n3 = \frac {2!\times (n-3)!}{(n-1)!}\times \frac {n!}{3!\times (n-3)!}=\frac n3$$
Here we used the well-known identity $$\sum_{i=0}^{n-1}\binom ik=\binom {n}{k+1}$$ which can easily be demonstrated inductively or combinatorially, See this reference