Assignment 10: Stochastic Processes Probability Theory and Stochastic Processes The method by which you arrive at an answer is as important as the answer itself. Therefore, a portion of your grade for...

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Assignment 10: Stochastic Processes Probability Theory and Stochastic Processes The method by which you arrive at an answer is as important as the answer itself. Therefore, a portion of your grade for the homework is based on showing your work. It is not sufficient to simply provide the answer. Further, if you make an assumption, clearly state it. 1. (10 points) Let B[n] be a Bernoulli random sequence equally likely taking on values [-1, +1]. Define the random process ( )0( ) sin 2 [ ] for 12X t p f t B n nT t n T ππ = + ≤ < +="" ="" ="" ="" a.="" determine="" the="" mean="" function="" µx(t)="" b.="" determine="" the="" covariance="" function="" kxx(t1,t2)="" 2.="" (10="" points)="" the="" inhomogenous="" poisson="" counting="" process="" n(t)="" is="" defined="" for="" t≥0="" as="" follows="" (a)="" n(0)="0" (b)="" n(t)="" has="" independent="" increments="" (c)="" for="" all="" t2="">t1, 2 2 1 1 2 1 ( ) [ ( ) ( ] exp ( ) , 0 ! nt t t t v dv P N t N t v dv n n λ λ        − = − >     ∫ ∫ λ(t) is called the intensity function. Note that it is a function of time for Inhomogeneous Poissson Counting Processes. Compare this with a Uniform Poisson Counting Process where it is a constant and find a. Its mean function mean function µN(t) b. Its correlation function RNN(t1,t2) 3. (10 points) Let W1(t) and W2(t) be two Wiener processes that are independent of one another with both defined for t≥0 with variance parameters α1 and α2, respectively. Define the random process given by X(t) = W1(t) – W2(t). a. What is RXX(t1,t2) ? b. What is the PDF fx(x;t)? 4. (10 points) Let W(t) be a standard Wiener process. Define the random process. X(t) = W2(t) a. Find the probability density fX(x;t). b. Find the conditional density fX(x2|x1;t2,t1) Assignment 10: Stochastic Processes Probability Theory and Stochastic Processes