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CS314 Numerical Methods

Homework must be submitted via Blackboard in PDF file format. The PDF file (i.e. the main file) should include Matlab code (if necessary) and also the Matlab code should be uploaded in separate files (i.e., .m files).

Your PDF file should be named as follows: FistnameLastnameHWxx.pdf, where xx repre- sents the homework number, e.g., 01.

You should write up your own solution, and you are not permitted to share or copy some- one else’s written solution or code. All projects are individual projects.

1. (10 points) For this problem we are going to implement our own Monte Carlo simulation of a web surfer using a network of six nodes represented by the following adjacency graph.

That is, gij = 1 if there is a outgoing link from node i to node j, and 0 otherwise. You may assume that just like in the original formulation of PageRank that 85% of the time a surfer follows a link on the web page being visited and that the other 15% they visit a random page. Compute the probability of landing on each page from up to 5000 simulations of our surfer’s movements on the network above. Also, write down the transition matrix M = [mij]6×6. The transition matrix M is referred as Google matrix (see, Page 332 of the textbook). Note that the sum of elements in each row is equal to 1.

Note: The sample code in the textbook should be a good starting point for this problem.

2. (10 points) Using the same network and probabilities as above, compute the probabilities of landing on each page using multiple surfers. You can do this as follows.

• Initialize a counter for every node.

• Initialize a surfer for every node.

• Have the surfers move to a different node with the probabilities following the transition matrix

M calculated in Problem 1.

• When the surfers move to a different node increment the counter that corresponds to that node by the amount of surfers that land on that node.

• Repeat this process 500 times. Compute the probabilities by dividing the count of each node by the number of simulations carried out.

Do your results agree with the previous problem?

3. (10 points) A famous probability puzzle is the Monty Hall problem. A contestant on a game show is given the choice of choosing between three doors. If they choose the correct door they win a car, otherwise nothing. Behind one door is the car, and behind the other two doors are goats. However, when you pick a door, the host, who knows what’s behind each of the doors, opens a door that you did not pick that contains a goat. The host then gives you the option to stay or switch. Write MATLAB code that simulates the Monty Hall problem. It should output the probability of success if you switch doors as well as the probability of success if you decide to stay. Run the code 1000 times. Do the results converge to what your intuition expected them to?

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关键词一关键词二关键词三关键词四关键词五和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 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