hw8_sol_sr - EE351K Probability, Statistics and Random...

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EE351K Probability, Statistics and Random Processes Spring 2008 Instructor: Vishwanath { sriram } @ece.utexas.edu Homework 8: Solutions Problem 1 The count of students dropping the course Probability and Stochastic Processes is known to be a Poisson process of rate 0.1 drops per day. Starting with day 0, the first day of the semester, let D ( t ) denote the number of students that have dropped after t days. What is p D ( t ) ( d ) ? Solution: D(t): Poisson process with the parameter λ = 0 . 1 . p D ( t ) ( d ) = ( λt ) d d ! e - λd = (0 . 1 t ) d d ! e - 0 . 1 d . Problem 2 For an arbitrary constant a , let Y ( t ) = X ( t + a ) . If X ( t ) is a stationary random process, is Y ( t ) stationary? Solution: X ( t ) is stationary. Thus, F X ( t 1 ) ,...,X ( t k ) ( x 1 ,...,x k ) = F X ( t 1 + τ ) ,...,X ( t k + τ ) ( x 1 ,...,x k ) , for all time shifts τ , all k , and all choices of sample times t 1 ,...,t k . Then,
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hw8_sol_sr - EE351K Probability, Statistics and Random...

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