This is intended to provide a set of self-contained examples, much like the Mudd math fun facts which demonstrate mathematics through biological examples.
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Average of a geometric distribution (extended to exponential) #19
If $p$ is the probability of dying in a day, expected lifetime is
(pN + 2 p(1-p)N + 3p(1-p)^2 N + …)/N
= p( 1 + 2(1-p) + 3(1-p)^2 + …)
= p(1 + 2q + 3q^2+ …). [ q=1-p]
= p (d/dq)(q+q^2+q^3+…)
= p (d/dq) ( q/(1-q))
= p (-d/dp) (1-p)/p)
= p (-d/dp) (1/p) -1
= p (1/p^2)+0
=1/p
Present this by deriving the sum 1 + 2q + 3q^2 + … first.
If $p$ is the probability of dying in a day, expected lifetime is (pN + 2 p(1-p)N + 3p(1-p)^2 N + …)/N = p( 1 + 2(1-p) + 3(1-p)^2 + …) = p(1 + 2q + 3q^2+ …). [ q=1-p] = p (d/dq)(q+q^2+q^3+…) = p (d/dq) ( q/(1-q)) = p (-d/dp) (1-p)/p) = p (-d/dp) (1/p) -1 = p (1/p^2)+0 =1/p
Present this by deriving the sum 1 + 2q + 3q^2 + … first.