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December 3rd, 2016, 03:57 AM   #1
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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)?
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December 3rd, 2016, 08:06 AM   #2
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Sum (2) is over all subscripts. Sum (3) separates that into a sum over all odd subscripts and a sum over all even subscripts.
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December 10th, 2016, 02:21 AM   #3
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Thanks, that make sense.
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