Consider an unfair coin with probability p of heads. n has been tossed (a deterministic number) n times. Denote by X ber of heads among the n tosses and by Y the number of tails. Show and Y are dependent. n has been tossed (a random number) N times where N ~ Poisson(A) by X the number of heads among the N tosses and by Y the number
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- Show that if X is not a deterministic random variable, then H(X) is strictly positive. What happens to the probabilities if a random variable is non-deterministic?1.Random variables are outcomes of deterministic experiments. Yes Incorrect Could be AlwaysSuppose that X k is a time-homogenous Markov chain. Show thatP{X3= j3, X2= j2|X0= j0,X1 = j1}= P{X3 = j3 | X2 = j2} P{X2 = j2|X1 = j1}.
- 57. Consider a situation where there is a cost that iseither incurred or not. It is incurred only if the valueof some random input is less than a specified cutoffvalue. Why might a simulation of this situation give avery different average value of the cost incurred thana deterministic model that treats the random input asfixed at its mean? What does this have to do with the“flaw of averages”?8. List the Gauss–Markov conditions required for applying a t & F-tests.Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3. Find limn→∞ P(Xn = 1).
- Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3.(a) Find P(X2 = 1).(b) Find the conditional probability P(X2 = 1|X1 = 1).(c) Find the conditional probability P(X1 = 1|X2 = 1).(d) Find limn→∞ P(Xn = 1).In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 5 minutes and that the checkout time per customer is also exponential with mean 10 minutes. Suppose further that will add a fourth counter. Counters 1,2, and 3 will open based on increments of two customers and counter 4 will open when there are 7 or more in the store. (a) The steady-state probabilities, for all . (b) The probability that a fourth counter will be needed. (c) The average number of idle counters.If Kt = B2t - t, where B is standard Brownian Motion, show that Kt is a martingale, and a markov process
- For the attached transition probability matrix for a Markov chain with {Xn ; n = 0, 1, 2,.........}: a) How many classes exist, and which two states are the absorption states? b) What is the limn->inf P{Xn = 3 | X0 = 3}? b) What is the limn->inf P{Xn = 1 | X0 = 3}?what is the step by step basis taken to solve the following jump discontinuity?Construct the Markov diagram for this game and show that the associate transition matrix T can be written as the following. Show that λ = 1 is an eigenvalue of T. If can answer both (a)and(b), else answer (b).