1. Aircraft arrive at the maintenance center according to a Poisson process N(t) with pa- rameter...
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1. Aircraft arrive at the maintenance center according to a Poisson process N(t) with pa- rameter A. In every aircraft a random number of parts is replaced according to a Poisson distribution with parameter 1. The numbers of parts replaced in different airplanes are independent. Denote by X(t) the total number of parts replaced until time t (assume that the shop maintains an infinite number of parts). (a) Is X(t) Poisson? (b) Compute Pr{X (t) = j | N(t) = n}. (c) Express Pr{X (t) = j} using an expectation on a function of N(t). 2. Let N(t) be a Poisson process with parameter A. (a) Assume that at a certain t > 0 in a Poisson process there is no arrival. Denote by W (t) the time interval until the next arrival. Show that the distribution of W(t) does not depend on t. How is W(t) distributed? (b) Let T be the time untl the first arrival. For 0 < T 4. Let W(t) be a Wiener process. Define a new process as: X(t) W (t)². Show that X(t) is Markov. Hint: use the following stages: (a) Define Y (t) sgn(W (t)). Show using total probability that: Pr{X(tn) 4. Let W(t) be a Wiener process. Define a new process as: X(t) W (t)². Show that X(t) is Markov. Hint: use the following stages: (a) Define Y (t) sgn(W (t)). Show using total probability that: Pr{X(tn) 5. Let T > 0 be an arbitrary number and let X(t) be a process that satisfies: Hx(t + mT) = Hx(t) Cx(tı + mT, t2 + mT) = Cx(tı,t2) for any integer m (this process is called Wide sense cyclostationary process). Define Y (t) 4 X(t + e), where e - U[0, T] (uniformly distributed between 0 and T) and independent of X(t). (a) Show that Y (t) is WSS. (b) Assume that, in addition, X(t) satisfies: Fx(tı+mT).X(t2+mT).X(ta +mT) (#1, 2, ..., In) = Fx(t1).x(2).. (t,) (T1, 12, ... , "n) Show that in this case Y (t) is stationary. Hint: use the characteristic function. 6. Let X(t) = A cos t+ B sin t, where A, B are i.i.d. random variables with zero mean, finite variance and finite and non-zero third moment. (a) Show that X (t) is WSS. (b) Show that X(t) is not stationary. Hint: use higher moments. 7. Which, of the following functions, cannot serve as autocorrelation functions of WSS pro- cesses: (a) R(T) = cos T (b) R(T) = sin T J1, I기 <1 10, I기> 1 (c) R(T) = (d) R(T) = |T|e-1r| %3D 1, T = 0 T+0 (e) R(r) = sin T 1. Aircraft arrive at the maintenance center according to a Poisson process N(t) with pa- rameter A. In every aircraft a random number of parts is replaced according to a Poisson distribution with parameter 1. The numbers of parts replaced in different airplanes are independent. Denote by X(t) the total number of parts replaced until time t (assume that the shop maintains an infinite number of parts). (a) Is X(t) Poisson? (b) Compute Pr{X (t) = j | N(t) = n}. (c) Express Pr{X (t) = j} using an expectation on a function of N(t). 2. Let N(t) be a Poisson process with parameter A. (a) Assume that at a certain t > 0 in a Poisson process there is no arrival. Denote by W (t) the time interval until the next arrival. Show that the distribution of W(t) does not depend on t. How is W(t) distributed? (b) Let T be the time untl the first arrival. For 0 < T 4. Let W(t) be a Wiener process. Define a new process as: X(t) W (t)². Show that X(t) is Markov. Hint: use the following stages: (a) Define Y (t) sgn(W (t)). Show using total probability that: Pr{X(tn) 4. Let W(t) be a Wiener process. Define a new process as: X(t) W (t)². Show that X(t) is Markov. Hint: use the following stages: (a) Define Y (t) sgn(W (t)). Show using total probability that: Pr{X(tn) 5. Let T > 0 be an arbitrary number and let X(t) be a process that satisfies: Hx(t + mT) = Hx(t) Cx(tı + mT, t2 + mT) = Cx(tı,t2) for any integer m (this process is called Wide sense cyclostationary process). Define Y (t) 4 X(t + e), where e - U[0, T] (uniformly distributed between 0 and T) and independent of X(t). (a) Show that Y (t) is WSS. (b) Assume that, in addition, X(t) satisfies: Fx(tı+mT).X(t2+mT).X(ta +mT) (#1, 2, ..., In) = Fx(t1).x(2).. (t,) (T1, 12, ... , "n) Show that in this case Y (t) is stationary. Hint: use the characteristic function. 6. Let X(t) = A cos t+ B sin t, where A, B are i.i.d. random variables with zero mean, finite variance and finite and non-zero third moment. (a) Show that X (t) is WSS. (b) Show that X(t) is not stationary. Hint: use higher moments. 7. Which, of the following functions, cannot serve as autocorrelation functions of WSS pro- cesses: (a) R(T) = cos T (b) R(T) = sin T J1, I기 <1 10, I기> 1 (c) R(T) = (d) R(T) = |T|e-1r| %3D 1, T = 0 T+0 (e) R(r) = sin T
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Related Book For
Introduction to Operations Research
ISBN: 978-1259162985
10th edition
Authors: Frederick S. Hillier, Gerald J. Lieberman
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