EE 560 Fall 2022 Final exam
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EE 560
Fall 2022 Final exam
(200 pts)
72 hour time limit. You may consult your notes, the textbook, and the materials on the course page, but outside resources are not permitted. No discussion of the exam or collaboration with other students is permitted. Complete all problems. Please contact me if you need any clarification about a problem or if you believe that you have found a typo or other error.
1. (50 pts) Let (Nt : t e R>0) be a Poisson process with rate λ . Let Xt = (t(t) - 1)^0 Nt and let Yt = Nt -Xt .
(a) (10 pts) Sketch a sample path of N and the corresponding sample paths for X and Y in the case
λ = 1.
(b) (10 pts) What are µN (t), RN (s, t), and CN (s, t)?
(c) (10 pts) Find µX (t), CX (s, t), and CN,X (s, t).
(d) (10 pts) Find µY (t) and CY (s, t).
(e) (10 pts) Which of N , X, and Y are mean-square differentiable? For which of the three processes
are the sample paths almost surely differentiable?
2. (50 pts) Let µX = 0 and RX (τ ) = 5e- |τ | - 4e-2|τ | + e-3|τ | . Justify all answers to parts of this question with computations.
(a) (10 pts) Using the frequency domain, show that there is a W.S.S. process X with these moment functions.
(b) (5 pts) Show that X is continuous.
(c) (10 pts) Show that X is continuously differentiable.
(d) (10 pts) Show that X is twice continuously differentiable.
(e) (10 pts) Find RXX \\ (0).
(f) (5 pts) Show that X mean ergodic.
3. (35 pts) Let X and Y be i.i.d. Exp(1) random variables. For 0 < t < 1, let Zt = min(X/(1 - t), Y/t). Thus Z0 = min(X, +o) = X and Z1 = min(+o, Y) = Y .
To analyze Z, the following transformation is useful. Let U = and let V = X +Y, so X = (1 -U)V and Y = UV . Observe that Zt = X/(1 - t) for 0 < t < U , Zt = Y/t for U < t < 1, and ZU = V .
(a) (5 pts) What distribution does Zt have?
(b) (5 pts) Find µZ (t).
(c) (10 pts) Find the joint p.d.f. of U and V , the p.d.f. of U , and the p.d.f. of V . What do you observe about these random variables?
(d) (15 pts) Find the correlation function RZ (s, t). (Assume that s < t, then condition on the events 0 < U < s, s < U < t, and t < U < 1.)
4. (65 pts) Let 0 < α < 1. Consider a discrete time Markov process (Xt : t e z) on states {-2, -1, 0, 1, 2} with one-step transition probability matrix
╱.0(0) P = .(.) 4(1) .α
The transition diagram is below. |
α 0 1 4 0 0 |
1 - α 1 - α 0 1 - α 1 - α |
0 0 1 4 0 α |
α(0) 1 . 4 . . 0 . |
α
α
(a) (5 pts) Is the process aperiodic?
(b) (10 pts) Find the equilibrium distribution π for these transition rates.
(c) (5 pts) Now we will consider MMSE prediction of the process. Find E[Xt].
(d) (15 pts) Find E[Xt IXt - 1], E[Xt IXt -2], and E[Xt IXt - 1 , Xt -2].
(e) (15 pts) Find Cov(Xt ), Cov(Xt , Xt+1), and Cov(Xt , Xt+2). Hint: E[UV] = E[E[UV IU]] = E[UE[V IU]].
(f) (10 pts) Eˆ [Xt IXt - 1], Eˆ [Xt IXt -2], and Eˆ [Xt IXt - 1 , Xt -2].
(g) (5 pts) Explain the relationship between your answers to parts (d) and (f) and the fact the X is a Markov process.
2022-12-15