Drop bears are large canivorous Australian marsupials, somewhat similar to koala

Question

Drop bears are large canivorous Australian marsupials, somewhat similar to koalas, that are known to attack campers by dropping onto tents during the night. (a) Suppose on any given night, all drop bears only drop once, and succesfully land on a tent with probability p, independently of other nights and other drop bears. Using a Markov chain and first step analysis, calculate the expected time between successful landings for a drop bear. (b) How is your answer related to the geometric distribution?

Explanation / Answer

Expected time between successful landings of a drop bear = Sum of (all possible time gaps * prob. of that time gap)

=> E(TG) = (p^2) * [ 1*1 + 2*(1-p) + 3*(1-p)^2 + 4*(1-p)^3 + ... + so on ] <-- Nth term is N*(prob. of full time gap = N)

E(TG) = (p^2) * [ { 1 + (1-p) + (1-p)^2 + ... + so on } * { 1 + (1-p) + (1-p)^2 + ... + so on } ]

Sum of an infinite GP with r < 1 => a/(1-r); where a is starting term and r is multiplying factor of GP

=> E(TG) = (p^2) * (1/p)^2 = 1 [Answer]

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Geomteric distribution:

It is related to the geomteric distribution in the way that each term is actually the probability of each X value --> 1, 2, 3, ... , till infinity => And hence, the total sum is supposed to be (and is, as calculated) equal to 1

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