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Consider the following discrete time process:| at integer times, a marble arrives which can be one of m colors. Each color is equally likely; all events at different times are independent. As each marble arrives, it is placed into a bowl Unless there is a marble of that color already in the bowl. Then, that marble is removed from the bowl and the new marble does not go into the bowl. (The pair is placed in a separate bin for shipping). Let Gn denote the number of marbles currently in the bowl after the nth marble arrives. Compute (a) the expectation (average) of G,, say, E(G„). (b) lim, E(Gn). Hint: We have the Law of iterated expectations: E(X) E(X)+ E(Y) E(E(X | Y)), and E(X + Y) =

Consider the following discrete time process:| at integer times, a marble arrives which can be one of m colors. Each color is equally likely; all events at different times are independent. As each marble arrives, it is placed into a bowl Unless there is a marble of that color already in the bowl. Then, that marble is removed from the bowl and the new marble does not go into the bowl. (The pair is placed in a separate bin for shipping). Let Gn denote the number of marbles currently in the bowl after the nth marble arrives. Compute (a) the expectation (average) of G,, say, E(G„). (b) lim, E(Gn). Hint: We have the Law of iterated expectations: E(X) E(X)+ E(Y) E(E(X | Y)), and E(X + Y) =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following discrete time process:| at integer times, a marble arrives which can be one of m colors. Each color is equally likely; all events at different times are independent. As each marble arrives, it is placed into a bowl Unless there is a marble of that color already in the bowl. Then, that marble is removed from the bowl and the new marble does not go into the bowl. (The pair is placed in a separate bin for shipping). Let G, denote the number of marbles currently in the bowl after the nth marble arrives. Compute (a) the expectation (average) of Gn, say, E(G,). (b) lim,- E(Gn). Hint: We have the Law of iterated expectations: E(X) Е(X) + Е(Y) E(E(X | Y)), and E(X + Y)
Transcribed Image Text:Consider the following discrete time process:| at integer times, a marble arrives which can be one of m colors. Each color is equally likely; all events at different times are independent. As each marble arrives, it is placed into a bowl Unless there is a marble of that color already in the bowl. Then, that marble is removed from the bowl and the new marble does not go into the bowl. (The pair is placed in a separate bin for shipping). Let G, denote the number of marbles currently in the bowl after the nth marble arrives. Compute (a) the expectation (average) of Gn, say, E(G,). (b) lim,- E(Gn). Hint: We have the Law of iterated expectations: E(X) Е(X) + Е(Y) E(E(X | Y)), and E(X + Y)
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