Question

Autocorrelation of an X(t) random process is Rxx (t1, t2) = 4e-t-t2 This a Gaussian process with mean zero. a) [6p] Is this process wide sense stationary? Briefly explain. b) [9p] Calculate the probability P (X(2)> 1) using the Table at the cover. c) [10p] Calculate approximately the probability P(X(2) > X(4) + 1). Some useful relations 1. Var(X(t)) = E({€)) - (E(X(t))) 2. R(X(t)X(t) = ELX(t-)X(02)]| 3. Var(X(c) +X)) = Var( (t) ) + Var (X (t2) - 2Cov(X (t,), X (t2)) 4. Cor(X(t).X(t)) = E[XCt,)X(t2)] - E[X(t,)]E[X(t2)]| 5. Value ofa Gaussian process at any time is a Gaussian random variable. 6. Set of values of a Gaussian process an any set of times are jointly Gaussian random variables. 7. Sum of Gaussian warizables is also a Gaussian random variable (z)0 0.02275 (E)6 0.50000 2.0 0.0 0.01786 2.1 0.46017 0.1 2.2 0.01390 0.42074 0.2 2.3 0.01072 0.38209 0.3 2.4 0.00820 0.34458 0.4 2.5 0.00621 0.30854 0.5 2.6 0.00466 0.27125 0.6 2.7 0.00347 0.24196 0.7 2.8 0.00256 0.2186 0.8 2.9 0.00187 0.18406 0.9 3.0 0.00135 0.15866 1.0 3.1 0.00097 0.13567 1.1 3.2 0.00069 0.11507 1.2 3.3 0.09680 0.00018 1.3 3.4 0.08076 0.00034 1.4 3.5 0.06681 0.00023 1.5 0.05480 3.6 0.00016 1.6 3.7 0.04457 0.00011 1.7 0.03593 3.8 0.00007 1.8 0.02872 3.9 0.00005 1.9

Answer 1

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