final_2007

Each person who comes in buys a pizza there are n

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Unformatted text preview: a in a period of 15 minutes is a (nonnegative integer) random variable K with known moment generating function MK (s) = E[esK ]. Each person who comes in buys a pizza. There are n types of pizzas, and each person is equally likely to choose any type of pizza, independently of what anyone else chooses. 3 Give a formula, in terms of MK (·), for the expected number of different types of pizzas ordered. Solution: Let D be the number of types of pizza the chef has to prepare, and let M be the number of people to enter the pizzeria. Let X1 , . . . , Xn be the respective indicator variables of each pizza. Thus if at least one person orders pizza type i, then Xi = 1, otherwise Xi = 0. Note that D = X1 + · · · + Xn . Thus we have: E[D] = E[E[D|M ]] = E[E[X1 + · · · + Xn |M ]] = n · E[E[Xi |M ] � �M � � n−1 = n·E 1− n � � n − 1 �M � = n−n·E n (letting s = log((n − 1)/n)) = n − n · E[esM ] � � = n − n · MK log((n − 1)/n) . Problem 4: (13 points) Let S be the set of arrival times in a Poisson process on R (i.e., a process that has been running forever), with rate λ. Each arrival time in S is displaced by a random amount. The random displacement associated with each element of S is a random variable that takes values in a finite set. We assume that the random displacements associated with different arrivals are independent and identically distributed. Show that the resulting process (i.e., the process whose arrival times are the displaced points) is a Poisson process with rate λ. (We expect a proof consisting of a verbal argument, using known properties of Poisson processes; formulas are not needed.) Solution: Since each point is perturbed independently of all the others, we can con­ sider the perturbation as follows: Let the perturbation values be {v1 , . . . , vm }, which occur with respective probabilities pi . Then consider splitting our d-dimensional Pois­ son process according to the probabilities pi , into processes N1 , N2 , . . . , Nm . By the results...
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This document was uploaded on 03/19/2014 for the course EECS 6.436 at MIT.

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