Let $S_1, S_2, S_3, ...$ be a sequence of independent, identically distributed (iid) random veriables, each exponentially distributed with a mean of $\mu_S$ (hence $\sigma_S = \mu_S$).
Let $M_n = max(S(n)_{n=1+(l-1)p}^{n=1+lp}) - min(S(n)_{n=1+(l-1)p}^{n=1+lp})$ for $l=1,2,3,...$, i.e. you take $p$ chunks of the sequence $S(n)$ and evaluate the difference between the $max$ and the $min$ of it.
What would be the $\mu_M$ and $\sigma_M$?
Thanks in advance!
It is unclear to me whether each maximum and minimum is supposed to be over $p$ terms or $p+1$ terms.
Let's suppose it is over $p$ terms. Then I would have thought memorylessness would mean that M was distributed like the maximum of $p-1$ i.i.d. exponential random variables (wait for the first to happen, start the clock, and see what the clock says when all the others have happened), which is like summing exponential random variables with different rates.
So I would have thought that the expectation would be $$\mu_M = \left(1+\frac12+\frac13+\cdots+\frac{1}{p-1}\right)\mu_S $$ so involving the $p-1^\text{th}$ harmonic number and slightly more than $(\log_e(p-1) +\gamma)\mu_S$, and the standard deviation would be something similar but with squares and square roots $$\sigma_M = \sqrt{1+\frac1{2^2}+\frac1{3^2}+\cdots+\frac{1}{(p-1)^2}}\;\sigma_S$$ which is bounded above by $\dfrac{\pi}{\sqrt{6}}\sigma_S$