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Includes an entropy exercise for independent three-sided dice and a Markov chain steady-state calculation for a car moving among broken, working, and shop states. Additional sections test martingale and stopping-time facts, and complete normal-moment calculations used in the derivation of the Black-Scholes formula, culminating in European call option pricing relationships.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/440-practice-final-exam-solution/364354/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/440-practice-final-exam-solution/364354.png","ImageObject",300,407,{"name":92,"@type":93},"Fahsai","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-09-23",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you show that the moment generating functions of Tn converge to the standard normal MGF?","Question",{"text":112,"@type":113},"Assume X has mean 0 and variance 1, then express MTn(t) in terms of MX(t/pn). Take logs to get LTn(t)=nLX(t/pn), use Taylor expansion around 0 for LTn(t), and compare with the log MGF of a standard normal.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the relationship between the strong law of large numbers and the weak law of large numbers?",{"text":117,"@type":113},"The strong law implies almost sure convergence of sample averages to the mean, which in turn guarantees convergence in probability. Convergence in probability yields the weak law.",{"name":119,"@type":110,"acceptedAnswer":120},"How are the normal-moment calculations used to connect to the Black-Scholes formula?",{"text":121,"@type":113},"Compute expectations of functions of a normally distributed variable, then apply these results to the log-normal dynamics under the risk-neutral measure. The derived expectations provide the terms needed to express the European call price C(K) using the normal cumulative distribution.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},364354,1791449146,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":29,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144,"preview_page_count":4},549768702563,"https://ap-avatar.wpscdn.com/avatar/8000c4aa63b76e948b?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786536092046926083","18.440 Practice Final Exam: 100 points Carefully and clearly show your work on each problem (without  \nwriting anything that is technically not true) and put a box around each of your 􀀌nal computations. This practice exam deals only with the portion of the course after the second midterm, while the actual 􀀌nal exam will cover the entire course.  \nI. (20 points) Let X be a random variable with 􀀌nite mean 􀀖 and variance 􀀛 2 . The central limit theorem states that if Xi are independent instances of X, then the quantities  \nTn := P~~1~~􀀛Xpin􀀀~~ ~~n􀀖  \nconverge in law to a standard normal random variable. That is  \nnl PfTn \u003C ag = Z􀀀 p12􀀙 e􀀀x2 =2dx:  \nProve, using the following steps, that the moment generating functions of the Tn (assuming they exist and are well de􀀌ned) converge to the moment generating function of a standard normal random variable.  \n1. Explain why it su􀀎ces to consider mean zero, variance one random variables X . You may then assume below that 􀀖 = 0 and 􀀛 2 = 1 .  \n2. Let MX(t) = EetX . Show that MTn(t) = MX(t= pn)n. Show also that M00X(0) = 1 and M0X(0) = 0 .  \n3. Let L X(t) = log MX(t) and show that LTn(t) = nL X(t= pn) .  \n4. Show that L0X(0) = 0 and L00X(0) = 􀀛 2 = 1 .  \n5. Show that if N is a standard normal random variable then LN(t) = t2 =2.  \n6. Use Taylor approximation to show that, for each 􀀌xed t, limn!1 LTn(t) = LN(t) . [Recall that Taylor approximation states that if R is twice di􀀋erentiable at zero then  \nR (t) = R(0) + R0 (0)t + R00 (0)t2 =2 + o(t2 ), for small t.]  \nSolution: Check the central limit theorem derivation in the textbook and slides.  \n1  \nII. (15 points) State the strong and weak laws of large numbers and explain why the strong law of large numbers implies the weak law of large numbers.  \nSolution: Check the law of large number explanations in the textbook and slides.  \nIII. (10 points) Let X and Y be the outcomes of independent die rolls, and let Z = X + Y. (Assume that these are 3-sided dice, taking the values 1, 2, and 3 with equal probability.) Compute the following:  \n1. The entropies H (X), H (Y ), H (Z), and H (X; Y ) .  \n2. Show that H (X; Y ) = H(X; Z) .  \n3. Verify by explicitly calculating both sides that H (X; Z) = H(Z) + HZ(X) .  \nSolution:  \n1. H (X) = H(Y ) = 􀀀 log ~~1~~3 = log 3 and H (X; Y ) = H(X) + H(Y ) = 2 log 3 .  \n2. Like the pair (X; Y ), the pair (X; Z) takes 9 values, all with equal probability. So H (X; Z) = 􀀀 log ~~1~~9 = 2 log 3 .  \n3. The variable Z takes 5 values: 2 ; 3 ; 4 ; 5 and 6 with probabilities ~~1~~9 , ~~2~~9 , ~~3~~9 , ~~2~~9 , and ~~1~~9 . Now,  \n6  \nHZ(X) =XP fZ = jgHZ =j (X)  \nj=2  \n= 19 log 1 + 29 log 2 + 39 log 3 + 29 log 2 + 19 log 1  \n4  1  \n= log 2 + log 3 :  \n9 3  \n2  \nAnd  \nH (Z) = 19 (􀀀 log 19) + 29 (􀀀 log 29) + 39 (􀀀 log 39) + 29 (􀀀 log 29) + 19 (􀀀 log 19)  \n4  4  1 = log 3 + (2 log 3 􀀀 log 2) + log 3  \n9 9 3 = 53 log 3 􀀀 49 log 2 :  \nSo indeed H (Z) + HZ(X) = 2 log 3 = H(X; Z) .  \nIV. (10 points) Elaine's trusty old car has three states: broken (in Elaine's possession), working (in Elaine's possession), and in the shop. Denote these states B, W, and S.  \n1. Each morning the car starts out B, it has a .5 chance of staying Band a .5 chance of switching to S by the next morning.  \n2. Each morning the car starts out S, it has a .75 chance of staying Sand a .25 chance of switching to W by the next morning.  \n3. Each morning the car starts out W, it has a .75 chance of staying W, and a .25 chance of switching to B by the next morning.  \nOver the long term, on what percentage of days does the car start out in state W?  \nSolution: Ordering the states B; W; S, we may write the Markov chain  \nmatrix as  \nM = 0 :2:55 @ 0  \n0  \n:75  \n:25  \n:50 1 :  \n:75A  \nWe 􀀌nd the stationarity probability vector 􀀙 = (􀀙B ; 􀀙W ; 􀀙S ) = (:2; :4 ; :4)  \nby solving 􀀙M = 􀀙 (with components of 􀀙 summing to 1) . So 􀀙W = :4.  \nV. (20 points)  \n3  \n1. Let X1 ; X2 ; : : : be independent random variables with expectation one. In which of the cases below is the sequenc","cbCaigQUU6ghVXQN","https://ap.wps.com/l/cbCaigQUU6ghVXQN","pdf",139750,"English","# I. Central limit theorem via moment generating functions\n# II. Laws of large numbers (strong and weak)\n# III. Entropy for independent dice and derived variables\n# IV. Markov chain stationarity probability\n# V. Martingales and stopping times\n# VI. Normal-moment calculations for Black-Scholes\n# VII. European call option pricing with given strike function","[{\"question\":\"How do you show that the moment generating functions of Tn converge to the standard normal MGF?\",\"answer\":\"Assume X has mean 0 and variance 1, then express MTn(t) in terms of MX(t/pn). Take logs to get LTn(t)=nLX(t/pn), use Taylor expansion around 0 for LTn(t), and compare with the log MGF of a standard normal.\"},{\"question\":\"What is the relationship between the strong law of large numbers and the weak law of large numbers?\",\"answer\":\"The strong law implies almost sure convergence of sample averages to the mean, which in turn guarantees convergence in probability. Convergence in probability yields the weak law.\"},{\"question\":\"How are the normal-moment calculations used to connect to the Black-Scholes formula?\",\"answer\":\"Compute expectations of functions of a normally distributed variable, then apply these results to the log-normal dynamics under the risk-neutral measure. The derived expectations provide the terms needed to express the European call price C(K) using the normal cumulative distribution.\"}]","440 Practice Final Exam - Solution | PDF",1790157195,15]