*** Homework must be submitted via Blackboard in PDF file format. The PDF file (i.e. themain file) should include Matlab code (if necessary) and also the Matlab code should be uploaded in separate files (i.e., .mfiles).*** 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 individualprojects.1.(10points)ForthisproblemwearegoingtoimplementourownMonteCarlosimulationofa web surfer using a network of six nodes represented by the following adjacencygraph.Thatis,gij=1ifthereisaoutgoinglinkfromnodeitonodej,and0otherwise.Youmayassume 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 probabilityoflandingoneachpagefromupto5000simulationsofoursurfer’smovementsonthe 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 to1.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 asfollows.•Initialize a counter for everynode.•Initialize a surfer for everynode.•HavethesurfersmovetoadifferentnodewiththeprobabilitiesfollowingthetransitionmatrixM 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 thatnode.•Repeatthisprocess500times.Computetheprobabilitiesbydividingthecountofeachnode by the number of simulations carriedout.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 adoor 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 1000times.Dotheresultsconvergetowhatyourintuitionexpectedthemto?Note: This is also problem 3 in the book4.(10 points) Another famous probability puzzle is the birthday problem. The problem states thatinarandomlyselectedgroupofnpeople,atleasttwohavethesamebirthdaywithsomeprob- ability P. Assume a year has 365 days. Write MATLAB code to compute and plot the probability of a match from n = 2 to n = 100. For roughly what n can we almost guarantee that a pair of people has the samebirthday?Note: This is similar to problem 7 in thebook5.(10 points) Say someone offers you a game where a fair coin is tossed at each stage of the game. Starting at 2 dollars, for every heads that appears the amount of money in the pot is dou- bled. If the flip results in a tails you keep all the money that is accrued in the pot. For instance, if our coin flip is HHHT, the pot has accrued 8 dollars, and you win that 8 dollars due to thefinal flip being a tails. How much money would you pay to have a chance at playing this game? Write MATLAB code to simulate playing this game 10000 times and plot the results of your earnings.Computetheexpectedvalueofplayingthisgame.Howmuchwouldyoupaytoplayitnow?Does your simulation agree with yourcomputation?6.(5 points) Let X and Y be random variables with finite expected values.Show that the following aretrue.(a)E(X+Y)=E(X)+E(Y)(b)E(cX) = cE(x), where c is aconstant.7.(5 points) Let X be the result of rolling a fair die. Compute thefollowing.(a)Expected value ofX.(b)The variance ofX.(c)The standard deviation ofX.8.(5 points) Is E[X2] equivalent to E[X]2? Why or why not? Justify yourreasoning.代写CS Finance|建模|代码|系统|报告|考试编程类：C代写,JAVA代写,数据库代写,WEB代写,Python代写,Matlab代写,GO语言,R代写金融类：统计，计量，风险投资，金融工程，R语言，Python语言，Matlab，建立模型，数据分析，数据处理服务类：Lab/Assignment/Project/Course/Qzui/Midterm/Final/Exam/Test帮助代写代考辅导天才写手,代写CS,代写finance,代写statistics,考试助攻E-mail:[email protected]微信：BadGeniuscs 工作时间：无休息工作日-早上8点到凌晨3点如果您用的手机请先保存二维码到手机里面，识别图中二维码。如果用电脑，直接掏出手机果断扫描。

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