Solution 1 EENG635 Probability and Stochastic Processes Homework 04 Due on Wednesday April 08 2015 This homework carries 5 points towards your nal grade
Solution EENG Probability and Stochastic Processes Homework Due on Wednesday April This homework carries points
Solution EENG Probability and Stochastic Processes Homework Due on Wednesday April
and Stochastic Processes Homework Due on Wednesday April This homework carries points towards your nal grade
Solution EENG Probability and Stochastic Processes Homework Due on
Wednesday April This homework carries points towards your nal grade
Solution EENG Probability and Stochastic Processes Homework
Solution EENG Probability
(Solution) 1 EENG635: Probability and Stochastic Processes Homework #04: Due on Wednesday, April 08, 2015 This homework carries 5 points towards your nal grade...

Category: General
Words: 1050
Amount: $12
Writer:

Paper instructions

Hi i want everything solved on paper1 EENG635: Probability and Stochastic Processes Homework #04 : Due on Wednesday, April 08, 2015 This homework carries 5 points towards your fnal grade 1) (5 points) Consider a random vector, X = b X 1 X 2 X 3 ··· X n B T . Let E ( X i ) = ? i and let ? = b ? 1 ? 2 ? 3 ··· ? n B T . Let C = [ c ij ] 1 ? i ? n 1 ? j ? n , be the co-variance matrix, where c ij = E [( X i - ? i )( X j - ? j )] and c ii = V ar ( X i ) = ? 2 i . Let ? = [ ? ij ] 1 ? i ? n 1 ? j ? n be the correlation-coeFfcient matrix, where ? ij = c ij ? i ? j and ? ii = 1 , ? i . Show that ? is positive semi-defnite. 2) (10 points) Consider a random vector, X = b X 1 X 2 X 3 ··· X n B T . Let E ( X i ) = ? i and let ? = b ? 1 ? 2 ? 3 ··· ? n B T . Let C = [ c ij ] 1 ? i ? n 1 ? j ? n , where c ij = E [( X i - ? i )( X j - ? j )] and c ii = V ar ( X i ) = ? 2 i . Show that 1 n E b ( X - ? ) T C - 1 ( X - ? ) B = 1 . 3) Ms. Cooper wishes to attend a conFerence which would take place in Hotel Mariott or Hotel Hilton. She is not sure where the conFerence will take place but has to make her decision to book a room, immediately. The rooms in Hotel Mariott cost $ 180 per night and those in Hilton cost $ 150 per night. The taxi charge From Hotel Mariott to Hotel Hilton and vice-versa is $ 30 . The odds that the conFerence will be held in Hotel Mariott is 5 : 1 . The conFerence is a two-day event. So she needs to book a room For two nights. IF she stays in the wrong hotel, she needs to take a taxi From the hotel she is staying to the conFerence venue and back on day 1 and again book a taxi once on day 2 . Assume that the taxi charges to the airport From both hotels are same, $ 60 . a) (5 points) Which hotel should she book a room in, iF she wants to minimize her risk or expected cost? Explain your solution in detail. b) (10 points) IF she decides which hotel she wants to reserve by ±ipping a Fair coin, with Pr { heads } = Pr { tails } = 1 2 , then show the risk set and the admissible rules. 4) (5 points) Consider a random variable, ? ? U (0 , 1) . It is desired to guess the value oF ? by making three observations oF a random variable, X 1 , X 2 and X 3 , where X i ? Bernoulli ( ? ) , i = 1 , 2 , 3 . Is S = X 1 +2 X 2 +3 X 3 6 a suFfcient statistic For ? ? JustiFy your answer ( Hint: See iF you can come up with two or more diFFerent combinations oF X 1 , X 2 and X 3 , that yield the same value oF S ). 5) (5 points) A communication system transmits 0 ’s or 1 ’s with equal probability. Whenever a zero is transmitted the channel corrupts the bit with noise which is a zero mean normal random variable with mean, ? 2 0 . IF a 1 is transmitted, then the channel corrupts the bit with noise which is a zero mean normal random variable with mean, ? 2 1 . The receiver makes n successive observations, X 1 , X 2 , ··· , X n and makes a decision on whether the transmitted bit was a 0 or a 1 . Show that S = ? n i =1 X 2 i is a suFfcient statistic For the receiver to make a decision. You can use the Fact that iF X i ? N (0 , ? 2 ) and X i are iid, then the pdF oF Z n = ? n i =1 X 2 i ? 2 , f z ( z, n ) , is given by f z ( z, n ) = 1 2 n 2 ? ( n 2 ) z n - 2 2 e - z 2 , where ?( n ) is the Gamma Function defned as ?( n ) = i ? x =0 x n - 1 e - x dx (iF n is an integer, ?( n ) = ( n - 1)! and ? p 1 2 P = ? ? ). 6) (10 points) Your company must make a sealed bid For a construction project. IF you succeed in winning the contract (by having the lowest bid), then you plan to pay another frm $ 100 , 000 to do the work. IF you believe that the minimum bid (in thousands oF dollars) oF the other participating companies can be modeled as the value oF a random variable that is uniFormly distributed on (70, 140), how much should you bid to maximize your expected proft? Explain all the steps in your solution in detail ( Hint: IF you want to win as well as make proft you need to bid at least $ 100 , 000 ).

Answer

Get Essay Answer
1,200,000+ Questions
Satisfaction guaranteed