Let X (t) be a Poisson counting process with arrival rate, λ. We form two related counting processes, Y1 (t) and Y2 (t), by randomly splitting the Poisson process, X (t). In random splitting, the i th arrival associated with X (t) will become an arrival in process Y1 (t) with probability p and will become an arrival in process Y2 (t) with probability 1 – p. That is, let S i be the i th arrival time of X (t) and define to be a sequence of IID Bernoulli random variables with Pr (Wi = 1) = p and Pr (Wi = 0) = 1 – p. Then the split processes are formed according to
Find the PMFs of the two split processes, PY1 (k; t) = Pr (Y1 (t) = k) and PY2 (k; t) = Pr (Y2 (t) = k). Are the split processes also Poisson processes?
Answer to relevant QuestionsDefine a random process according to X[n] = X [n– 1] + Wn , n = 1, 2, 3, … Where X  = 0 and Wn is a sequence of IID Bernoulli random variables with and Pr( Wn = 1)= p and Pr( Wn = 0) = 1 – p. (a) Find the PMF, PX ...In this problem, we develop an alternative derivation for the mean function of the shot noise process described in Section 8.7, Where the are the arrival times of a Poisson process with arrival rate, λ, and h (t) is an ...A random process X (t) consists of three- member functions: x1 (t) = 1 x2 (t) = – 3, and x3(t) = sin (2πt). Each member function occurs with equal probability. (a) Find the mean function, µX (t). (b) Find the ...The three letters C, A, and T represent the states of a word- generating system. Let the initial state probability vector be (1/ 3 1/ 3 1/ 3) for the three letters, respectively. The transition matrix is given as What is the ...Find the steady state probability distribution for the web search engine model of exercise 9.7. A Web Search Engine Model Suppose after we enter some keywords into our web search engine it finds five pages that contain ...
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