By William Skeen
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Example text
Then for \s\ < 1, H{s) = Proof: n(P(s)). 2B shows that m = Σ ^n{P{s)Y = π(Ρ(5)). f. , H(s) = Eis^") = E„E{^''\N = n) = E„Es'- = E^{Pis)Y = Σ = ms)). 1: The number of times Ν that an animal visits an area containing a trap is a geometric random variable with P{N = n} = (1 — 0)0" (n = 0 , 1 , 2 , . . ) , and the probability that the animal is caught in the trap when it visits the area is p. If the anhnal is released each time it is caught, find the probability distribution of 7, the total number of times it is caught.
Is distributed as X, Proof: These results can be obtained directly from the definition and the theory of conditional expectations. f's. f. of Sff is given by His) = π(Ρ(5)). t. 1) + n^'\P{s))P^^\s). 8A, ES^ = H^'\\) = n^'\\)P^'\l) var = (EiV)(EJ^), = H^^\l) + H^'\l) - \Η^^\\)Υ = π^'\\)ΐΡ^^\\) + ié^\\) + P<^>(1) + π^'\\) - {Ρ^'\\)Ϋ^ W'\\)Y^iP''\\)V = (EiV)(var Χ) + (var N){EXf. • There is a class of compound distributions that has been studied in some detail in probability theory. v. Feller (1968, pp.
Conditions (c) [ a special case of (b)] are useful in probability applications when the {a„}, {b„} are sequences of probabilities. In latter sections of this text we shall be interested in the limiting behavior of the c„. A useful theorem in this direction is the following. 4: If {c„} = {a„} . {f>„}, Ση=οα„ exists, and lim„^ ^ c„ exists. Then c„ = ( £ αΛ lim b„. 3 implies that lim^n (1 — s)B(s) = β and lim^ij (1 — s)C(s) = y. Now for \s\ < 1, C(s) = A{s)B{s) and lim(l - s)C(5) = lim A(s) lim(l - 5)^(5) = aß < 00 sTl sTl sTl we may conclude that Y = aß and the result follows.