My Math Forum Expectations

 December 3rd, 2016, 02:57 AM #1 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 Expectations $\displaystyle S_n$ where n>=0 represents a renewal $\displaystyle N(t)$ represents the number of renewals up to time t $\displaystyle E[N(t)] = \sum_{n=1}^{\infty} P(N(t) >= n )$ (1) $\displaystyle E[N(t)] = \sum_{n=1}^{\infty} P(S_n <= t )$ (2) $\displaystyle E[N(t)] = \sum_{n=0}^{\infty} P(S_{2n+1} <= t ) +\sum_{n=1}^{\infty} P(S_{2n} <= t )$ (3) Can anyone please explain to me how you go from (2) -> (3)?
 December 3rd, 2016, 07:06 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,159 Thanks: 866 Sum (2) is over all subscripts. Sum (3) separates that into a sum over all odd subscripts and a sum over all even subscripts.
 December 10th, 2016, 01:21 AM #3 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 Thanks, that make sense.

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