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22.8. Wald's equation. Let X₁, X₂,... be independent and identically distributed with finite mean, and put S=X₁ + +X Suppose that is a stopping time: 7 has positive integers as values and [T=n] Eo(X₁,..., X); see Section 7 for examples. Suppose also that E[T]<∞. T (a) Prove that (22.21) E[S,]=E[X₁]E[T]. - (b) Suppose that X, is +1 with probabilities p and q, pq, let 7 be the first n for which S, is a or b (a and b positive integers), and calculate E[r]. This gives the expected duration of the game in the gambler's ruin problem for unequal p and q. ...

22.8. Wald's equation. Let X₁, X₂,... be independent and identically distributed with finite mean, and put S=X₁ + +X Suppose that is a stopping time: 7 has positive integers as values and [T=n] Eo(X₁,..., X); see Section 7 for examples. Suppose also that E[T]<∞. T (a) Prove that (22.21) E[S,]=E[X₁]E[T]. - (b) Suppose that X, is +1 with probabilities p and q, pq, let 7 be the first n for which S, is a or b (a and b positive integers), and calculate E[r]. This gives the expected duration of the game in the gambler's ruin problem for unequal p and q. ...

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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22.8. Wald's equation. Let X₁, X2,... be independent and identically distributed with finite mean, and put S=X₁ + is a stopping +X Suppose that time: 7 has positive integers as values and [T=n] Eo(X₁,..., X); see Section 7 for examples. Suppose also that E[T]<∞. (a) Prove that (22.21) E[S,] E[X₁]E[T]. (b) Suppose that X, is +1 with probabilities p and q, pq, let 7 be the first n for which S, is a or b (a and b positive integers), and calculate E[r]. This gives the expected duration of the game in the gambler's ruin problem for unequal p and q.
Transcribed Image Text:22.8. Wald's equation. Let X₁, X2,... be independent and identically distributed with finite mean, and put S=X₁ + is a stopping +X Suppose that time: 7 has positive integers as values and [T=n] Eo(X₁,..., X); see Section 7 for examples. Suppose also that E[T]<∞. (a) Prove that (22.21) E[S,] E[X₁]E[T]. (b) Suppose that X, is +1 with probabilities p and q, pq, let 7 be the first n for which S, is a or b (a and b positive integers), and calculate E[r]. This gives the expected duration of the game in the gambler's ruin problem for unequal p and q.
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