ISE 538 Introduction to Probability Models
Final : All parts are worth 10 points.
概率模型代写 1.There are 2 urns. Urn 1 has 2 red and 3 blue balls, urn 2 has 4 red and 2 blue balls. At each stage a ball is randomly chosen from one of the urns,
1.
There are 2 urns. Urn 1 has 2 red and 3 blue balls, urn 2 has 4 red and 2 blue balls. At each stage a ball is randomly chosen from one of the urns, its color is noted and it is then replaced. If the chosen ball was red, then the next selection is made from the same urn with probability .6; it it is blue then the next selection is made from the same urn with probability.2. Let Xn be the urn from which the nth ball is withdrawn. Let Yn be 1 if the nth ball selected is red and 0 if it is blue.
(a) Is Xn, n ≥ 1 a Markov chain. If so, give its transition probability matrix.
(b) Is Yn, n ≥ 1 a Markov chain. If so, give its transition probability matrix.
(c) What fraction of selected balls is red.
2. 概率模型代写
Arrivals to a single server system only enter the system if they find the server idle. Service times are exponential with rate µ. Wet and dry arrival periods alternate. In a wet period, arrivals are according to a Poisson process with rate λ1; in a dry period, arrivals are according to a Poisson process with rate λ2. Wet periods last for an exponential time with rate α1and dry periods last for an exponential time with rate α2. Let the state of the system be (1, j) if we are currently in a wet period and there are j in the system, j = 0.1; and let it be (2, j) if we are currently in a dry period and there are j in the system.
(a) Give the transition diagram.
(b) Give the balance equations. Do not solve the equations. In terms of the long run proportions Pi,j , i = 1, 2, j = 0.1 find
(c) the average number of customers in the system;
(d) the proportion of arrivals that do not enter the system.
3. 概率模型代写
Consider the previous problem, and let T1 be the time until the next arrival if we are currently in a wet period, and let T2 be the time if we are currently in a dry period. If λ1 = 2, λ2 = 1, α1 = 2, α2 = 4. find E[T1] and E[T2].
4.
Let N(t), t ≥ 0 be a Poisson process with rate λ.
(a) Find P(N(6) − N(3) = 4|N(2) = 2).
(b) Find E[N(2)(N(6) − N(2))].
(c) Find E[N(2)N(6)].
5. 概率模型代写
A coin that comes up heads with probability p is flipped k times. If Y is the total number of heads in the k flips, then the coin is flipped an additional Y times. Let X be the total number of heads in all flips.
(a) What is E[X|Y = i].
(b) Find E[X].
(c) If k = 4, find P(X = 4).
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