Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of the n options being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul- tiple objects. For each integer k with 0 < k < n, define pn (k) to be the probability that a randomly-chosen bucket contains exactly k objects. By definition, pn(0) + pn (1) + Pn(2) +. . .+ Pn (n – 1) + Pn(n) = 1. (a) Determine each of the following probabilities: p4(0), p4(1), p4(2), p4(3), p4(4). Briefly explain how you computed each of these probabilities.

Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of the n options being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul- tiple objects. For each integer k with 0 < k < n, define pn (k) to be the probability that a randomly-chosen bucket contains exactly k objects. By definition, pn(0) + pn (1) + Pn(2) +. . .+ Pn (n – 1) + Pn(n) = 1. (a) Determine each of the following probabilities: p4(0), p4(1), p4(2), p4(3), p4(4). Briefly explain how you computed each of these probabilities.

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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Question

Let n be a positive integer. Suppose you have n objects and n buckets.
For each object, randomly place it into a bucket, with each of the n options being equally likely.
Through this random process, some buckets may end up empty, while other buckets may have mul-
tiple objects.
For each integer k with 0 < k < n, define pn (k) to be the probability that a randomly-chosen bucket
contains exactly k objects. By definition, pn(0) + pn (1) + Pn(2) +. . .+ Pn (n – 1) + Pn(n) = 1.
(a) Determine each of the following probabilities: p4(0), p4(1), p4(2), p4(3), p4(4).
Briefly explain how you computed each of these probabilities.
() -
(b) In class, we showed that pPn(k)
1
1- -
n-k
where
k
n!
k!(n – k)!"
n
Clearly explain why this formula is true for all integers n and k with 0 < k < n.

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Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0 < p < 1, of landing heads, is flipped. If heads appear, then a ball is chosen at random from the urn and is replaced by a white ball; if tails appear, then a ball is chosen from the urn and is replaced by a black ball. Let Xn denote the number of white balls in the urn after the n stages. Q6(i.) Is {Xn, n 2 0} a Markov Chain? If so, explain why. Q6(ii.) Compute the transition probabilities Pij: (Hint: At some stage, suppose there are i white balls (and N – i black balls). Think how you can get i or i – 1 or i +1 white balls in the next stage.) Q6(iii.) Let N 2. Then the states are 0, 1, 2 and the transition probability matrix is given by 를 + 올 끌 Find the proportion of the time in each state.