FINAL EXAM 07-08 - Z 1 W(t)dW(t Question 4(20 marks Suppose an urn contains 10 red balls and 5 white balls One ball is selected af-ter another

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FINAL EXAM FOR EE AND IT SCHOOLS INSTRUCTIONS: The following documents are allowed: Dictionaries, scienti±c calculators and copies of probability tables. Question 1.(20 marks). (a) De±nition of a second order stochastic process ? (b) De±nition of a weakly stationary process and its spectral representation ? Question 2.(20 marks). Let Xn, n = 0,²1,²2, . .. be a Gaussian process such that E(Xn) = 0, n = 0,²1,²2, . ... Prove that it is weakly stationary if and only if it is strictly stationary i.e. for any numbers k, h, n1, n2, . .., nk = 0,²1,²2, . .. we have (Xn1 ,Xn2 , . ..,Xk) d = (Xn1+h,Xn2+h, . ..,Xnk+h), where d = denotes the equality in distribution. Question 3(20 marks). (a) De±nition od a Brownian motion W(t), t [0, 1]? (b) Using de±nition of the Ito integral with respect to Brownian motion com- pute the following Ito integral
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Unformatted text preview: Z 1 W(t)dW(t). Question 4(20 marks). Suppose an urn contains 10 red balls and 5 white balls. One ball is selected af-ter another without replacements. Let {Xn, n = 1, 2, . .., 15} denote the colour of the nth ball that is selected. Show that the random process is not a Markov chain. Question 5(20 marks). A man either passes the Saigon bridge or the Binh trieu bridge to work each day. Suppose that he never passes the Saigon bridge two days in a row; but if he passes the Binh Trieu bridge one day, then the next day he is just as likely to pass the Binh trieu bridge again as he is to pass the Saigon bridge. (a) Describe the Markov chain and its transition matrix. (b) What is the percentage of time, in the long run, the man in question passes the Saigon bridge ?...
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This note was uploaded on 10/28/2009 for the course MATH ma024 taught by Professor Thu during the Spring '09 term at Vienna EBA.

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