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
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
Related questions
Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87ec14f1-3829-4fe0-9141-23f1282d8e99%2Fd23f188a-415a-453d-b0f4-781d50fe29c6%2Fnnpyum_processed.jpeg&w=3840&q=75)
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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