[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-393867-105":59,"doc-detail-393867-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","studytools-exams-sample-questions","Studytools Exams Sample Questions","","Practice exam questions covering core topics in probability and statistics, including branching processes and extinction probability, the gambler’s ruin problem with conditional outcomes from prior coin flips, exponential-family modeling where a Poisson mean-variance relationship is identified, and logistic regression model selection using Bayesian Information Criterion (BIC) from deviance and parameter counts. Additional material evaluates autoregressive time series stationarity via characteristic equations and roots.",{"@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/studytools-exams-sample-questions/393867/",{"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/studytools-exams-sample-questions/393867.png","ImageObject",300,407,{"name":92,"@type":93},"Cipher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-26",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How is the extinction probability computed in the branching-process question?","Question",{"text":112,"@type":113},"Compute the expected number of offspring and solve the associated quadratic equation for the smallest positive extinction probability. Then raise the single-member extinction probability to the power of the current population size (six), using independence.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"In the gambler’s ruin problem, how is the probability of eventual ruin found after 10 flips?",{"text":117,"@type":113},"Update the gambler’s chip count using the first 10 results, then use the fair-coin gambler’s ruin relationship to compare moving down before reaching the upper boundary. The resulting probability falls into the provided answer range.",{"name":119,"@type":110,"acceptedAnswer":120},"How does BIC determine the optimal logistic regression model in the policy claim question?",{"text":121,"@type":113},"For each model, compute BIC as Deviance plus p*ln(n), where p includes the intercept and n is 1,000. Select the model with the minimum BIC value.","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},393867,1790643939,{"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":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":52,"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},687208528416,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","1.  \nEach member of a population independently produces a random number of offspring with the following probabilities:  \n\n| Number of\u003Cbr>offspring | 0 | 1 | 2 | 3+ |\n| --- | --- | --- | --- | --- |\n| Probability | 0.375 | 0.125 | 0.500 | 0.000 |\n\nIn the current generation, there are six people, each of whom has the ability to produce offspring. Calculate the probability that this population will eventually go extinct.  \nA. Less than 0.2  \nB. At least 0 .2, but less than 0.4  \nC. At least 0 .4, but less than 0.6  \nD. At least 0 .6, but less than 0.8  \nE. At least 0.8  \n1. Solution  \nKey: A  \nSolution:  \nThe expected number of offspring, 􀟤 = ∑0 􀝆􀜲􀯝 = .125 + 2 ∗ .5 = 1.125 > 1. This implies that 􀟨0 \u003C 1, so that the family of each current member of the population will go extinct with probability:  \n∞  \n􀟨0 = 􀷍 􀟨􀯝0􀜲􀯝 = .375 + . 125􀟨0 + . 5􀟨20  \n􀯝 =0  \nThe solutions to this quadratic equation are { .75, 1}. According to Ross, 􀟨0 is the smallest positive value satisfying the above equation, so the extinction probability of each member is:  \n􀟨0 = 0.75.  \nSince there are currently six members, each with independent extinction probabilities, the population extinction probability is:  \n􀟨60 = 0.756 = 0.178  \nClassification: A.6.g – Branching Processes  \nText Reference: Ross, Chapter 4.7 (p 234)  \nItem Notes: Similar to Ross Example 4.33 & 4.34 (p 236-7), but with different values.  \n2.  \nYou are given the following information:  \n• A fair coin is flipped by a gambler with 20 chips at time 0.  \n• The gambler wins 1 chip if the coin turns “head”.  \n• The gambler loses 1 chip if the coin turns “tail”.  \n• The gambler will stop playing when he either has lost all of his chips or he reaches 60 chips.  \n• Of the first 10 flips, there are 7 “heads” and 3 “tails”.  \nCalculate the probability that he will lose all of his chips, given the results of the first 10 flips.  \nA. Less than 0.50  \nB. At least 0.50, but less than 0.65  \nC. At least 0 .65, but less than 0.80  \nD. At least 0 . 80, but less than 0.95  \nE. At least 0.95  \n2. Solution  \nKey: B  \nSolution:  \nSince we are told that the coin is fair, the probability of winning on any given toss is 1/2 . After the first 10 flips, he had 24 chips (20+7-3) .  \nThe probability of going down 24 chips before going up another 36 chips is:  \n36  \n36 + 24  \n= 0.60  \nClassification: A.6.f– Gamblers ruin problem  \nText Reference: Ross 4.5 Item Notes:  \n3.  \nYou are given:  \n• A response variable 􀜻 is in the exponential family,􀝂 (􀝕) = 􀜿(􀝕, 􀟶) ∗ exp[􀝕􀟠~~ ~~−􀟶􀜽(􀟠)]  \n• 􀜽(􀟠) = 􀝁 􀰏  \n• 􀟤 = 􀜧 (􀜻)  \n• Dispersion parameter, 􀟶 = 1  \nDetermine an expression for 􀜸􀜽􀝎(􀜻) as a function of 􀟤 .  \nA. 􀟤 2  \nB. 􀟤  \nC. 1/􀟤  \nD. 1  \nE. 􀝁 􀰓  \n3. Solution  \nSolution: B  \n􀜸􀜽􀝎(􀝕) =  (􀟠) = 􀝁 􀰏  \n􀟤 =  (􀟠) = 􀝁 􀰏  \n􀃎 􀜸􀜽􀝎(􀝕) = 􀟤  \nAlternatively, recognize (e.g. Table 3.1 p 36 de Jong) from the given information that 􀜻 is Poisson(􀟤) from which it follows that variance is also 􀟤 .  \nClassification: C.1.a – Understand the relationship between mean and variance by model family member  \nText Reference: de Jong, Chapter 3.1 Item Notes:  \n4.  \nAn actuary wants to estimate the probability of a home insurance policy having a claim by using a logistic regression model. He has the following pieces of information from 1,000 historical policies:  \n• Cost of the home, in $000s  \n• Age of the home, in years  \n• Whether or not there was a claim on the policy  \nThe actuary is considering a number of different model specifications. Below are the models he is considering along with the calculated deviance of each model:  \n\n| Model \\# | Included Variables | Deviance |\n| --- | --- | --- |\n| 1 | Intercept + Cost | 1085.0 |\n| 2 | Intercept + Cost + Age | 1084.8 |\n| 3 | Intercept + Cost + (Cost * Age) | 1083.0 |\n| 4 | Intercept + Cost + Cost2 + Cost3 | 1081.9 |\n| 5 | Intercept + Cost + Cost2 + Cost3 + Cost4 | 1081.6 |\n\nDetermine the optimal model using the Bayesian Information Criterion.  \nA. 1  \nB. 2  \nC. 3  \nD. 4  \nE. 5  \n4. Solution  \nKey: A  \nSolution:  \nThe","cbCaijG2bKkW3e40","https://ap.wps.com/l/cbCaijG2bKkW3e40","pdf",119168,"English","# Questions\n## Probability and Markov-type processes\n## Exponential family and GLM modeling\n## Logistic regression model selection\n## Autoregressive time series","[{\"question\":\"How is the extinction probability computed in the branching-process question?\",\"answer\":\"Compute the expected number of offspring and solve the associated quadratic equation for the smallest positive extinction probability. Then raise the single-member extinction probability to the power of the current population size (six), using independence.\"},{\"question\":\"In the gambler’s ruin problem, how is the probability of eventual ruin found after 10 flips?\",\"answer\":\"Update the gambler’s chip count using the first 10 results, then use the fair-coin gambler’s ruin relationship to compare moving down before reaching the upper boundary. The resulting probability falls into the provided answer range.\"},{\"question\":\"How does BIC determine the optimal logistic regression model in the policy claim question?\",\"answer\":\"For each model, compute BIC as Deviance plus p*ln(n), where p includes the intercept and n is 1,000. Select the model with the minimum BIC value.\"}]","Studytools Exams Sample Questions | PDF",1790395130,25]