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December 3rd, 2016, 03:57 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, 08:06 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,492 Thanks: 632 
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, 02:21 AM  #3 
Senior Member Joined: Feb 2015 From: london Posts: 121 Thanks: 0 
Thanks, that make sense.


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