A sequence of random variables, Xn, is to be approximated by a straight line using the estimate, Ẋ n = a+ bn. Determine the least squares (i. e., minimum mean squared error) estimates for a and b if samples of the sequence are observed.
Answer to relevant Questions(a) Prove that any sequence that converges in the mean square sense must also converge in probability. Use Markov’s inequality. (b) Prove by counterexample that convergence in probability does not necessarily imply ...Let X (t) = A(t) cos (ω0t + θ), where A(t) is a wide sense stationary random process independent of θ and let θ be a random variable distributed uniformly over . Define a related process Y (t) = A (t) cos((ω0 +ω1) t + ...Let Wn be an IID sequence of zero- mean Gaussian random variables with variance . Define a discrete- time random process, X[ n] = pX[ n – 1]+ Wn, n = 1, 2, 3, … where X[ 0] = W0 and is a constant. (a) Find the mean ...Suppose X (t) is a Weiner process with diffusion parameter λ = 1 as described in Section 8.5. (a) Write the joint PDF of X1 = X (t1) and X2 = X (t2) for t2 < t1 by evaluating the covariance matrix of X = [X1, X2] T and ...Suppose the arrival of calls at a switchboard is modeled as a Poisson process with the rate of calls per minute being λ a = 0.1 (a) What is the probability that the number of calls arriving in a 10- minute interval is less ...
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