[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-398877-105":59,"doc-detail-398877-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","mas-i-sample-questions-inversion-method-spline-knots-resampling-regression-splines","MAS I Sample Questions - Inversion Method, Spline Knots, Resampling, Regression Splines","","Sample exam questions and solutions covering Monte Carlo simulation using the inversion method for a Gamma distribution, identification of bias–variance–error relationships when selecting spline knot counts, and evaluation of resampling statements for LOOCV versus k-fold cross-validation. Additional items compute total regression spline coefficients from given spline knot counts, spline degrees, and an intercept term. Includes answer choices, keyed solutions, brief reasoning, and references to standard texts and classification labels.",{"@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/mas-i-sample-questions-inversion-method-spline-knots-resampling-regression-splines/398877/",{"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/mas-i-sample-questions-inversion-method-spline-knots-resampling-regression-splines/398877.png","ImageObject",300,407,{"name":92,"@type":93},"mieayamfan","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you simulate a Gamma random variable using the inversion method in the given problem?","Question",{"text":112,"@type":113},"Treat the Gamma(2, 1) variable as the sum of two independent exponential variables with rate 1, then use the inversion step −ln(U1) − ln(U2) with the provided uniform draws.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Which statistic corresponds to each line when selecting the number of knots for a cubic spline?",{"text":117,"@type":113},"Use the expected behavior: more knots increase flexibility and variance (line Y), while squared bias and training MSE decrease with flexibility (lines X and Z). Test MSE follows a U-shape (line W), and only the combination matching these conditions is correct.",{"name":119,"@type":110,"acceptedAnswer":120},"What are the correct statements comparing LOOCV and k-fold cross-validation?",{"text":121,"@type":113},"LOOCV is k-fold cross-validation with k = n (true). For k \u003C n, LOOCV does not have higher variance than k-fold (false). The validation set approach tends to overestimate test error more than LOOCV (false statement as written).","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},398877,1791288446,{"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":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"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":46,"preview_page_count":4},962090893677,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Exam MAS I Sample Questions (as of 11/21/2017)  \n1.  \nAn actuary is using the inversion method to simulate a gamma random variable with mean and variance both equal to 2. Two random draws from the uniform distribution (0,1) are independently made, and their values are 0.2 and 0.8.  \nCalculate the value of the simulated gamma random variable.  \nA. Less than 0.5  \nB. At least 0.5, but less than 1.0  \nC. At least 1.0, but less than 1.5  \nD. At least 1.5, but less than 2.0  \nE. At least 2.0  \n1. Solution  \nKey: D  \nSolution: By hypothesis, we know that the actuary is simulating a gamma random variable with 􀀡, 􀀡  = (2, 1) . Thus, it can be thought of the sum of two independent exponential random variables each with rate 1. Using the inversion method, we have:  \n− ln0.2 − ln 0.8 = 1.833.  \nClassification: A.8.c – Simulation – Inversion Method  \nText Reference: Ross, 11.2.1 (example 11.3) & 11.3.2  \nItem Notes: The author states that when simulating exponential random variables using either U or (1-U) as values is valid (example 11.3, p 650) . However in this question the simulated values are chosen such that the answer is the same either way.  \n2.  \nYou want to fit a cubic spline to a large dataset and need to determine the number of knots to use. Below is a chart of four statistics from this model valued for various numbers of knots:  \nDetermine which set of statistics below best describes each line.  \nA. W is Test MSE X is Variance Y is Squared Bias Z is Train MSE  \nB. W is Variance X is Squared Bias Y is Test MSE Z is Train MSE  \nC. W is Train MSE X is Test MSE Y is Variance Z is Squared Bias  \nD. W is Test MSE X is Train MSE Y is Variance Z is Squared Bias  \nE. W is Variance X is Train MSE Y is Test MSE Z is Squared Bias  \n2. Solution  \nKey: D  \nSolution:  \nAll else equal, a spline with more knots will be more flexible, and allow the model to better fit the training data.  \nVariance increases with flexibility à Line Y  \nBias squared and Train MSE both decrease with flexibility à Lines X&Z Test MSE generally exhibits a ‘U’ shaped behavior à Line W  \nà Only answer D meets these conditions  \nAlso Test MSE is the sum of Variance + Bias Squared + Irreducible Error, and only Line W can be the sum oftwo of the other values  \nClassification: C.2.q – Test vs Train Error  \nText Reference: James et al, p 29-37 Item Notes:  \n3.  \nYou are given the following statements about different resampling methods:  \nI. Leave-one-out cross-validation (LOOCV) is a special case ofk-fold cross-validation  \nII. k-fold cross-validation has higher variance than LOOCV when k \u003C n  \nIII. LOOCV tends to overestimate the test error rate in comparison to validation set approach Determine which of the above statements are correct.  \nA. I only  \nB. II only  \nC. III only  \nD. I, II, and III  \nE. The correct answer isn’t given by (A),(B),(C), or (D)  \n3. Solution  \nKey: A  \nSolution:  \nI. TRUE: LOOCV is just k-fold cross-validation where k = n  \nII. FALSE: LOOCV has higher variance than k-fold validation, for k \u003C n, because of the high degree of correlation in the training data sets.  \nIII. FALSE: Validation set approach tends to overestimate the test error rate more than LOOCV, which is one ofits drawbacks.  \nClassification: C.2.p – Cross Validation  \nText Reference: James et al, Chapter 5.1 Item Notes:  \n4.  \nYou are given the following models which contain regression splines:  \n\n| Model | Numbers of\u003Cbr>Spline Knots | Degree of\u003Cbr>Regression\u003Cbr>Spline |\n| --- | --- | --- |\n| A | 6 | 4 |\n| B | 5 | 5 |\n| C | 8 | 2 |\n| D | 10 | 3 |\n\nCalculate the total number of the regression coefficients in the four models.  \nA. Less than 5  \nB. At least 15, but less than 25  \nC. At least 25, but less than 35  \nD. At least 35, but less than 45  \nE. At least 45  \n4. Solution  \nKey: E  \nSolution:  \n\\# of coefficients = Degree of Regression Spline + Numbers of Spline Knots + Intercept  \nNumber of coefficients sum for Model A: 6 + 4 + 1 = 11  \nNumber of coefficients sum for Model B: 5 + ","cbCainQYEPlEJPfG","https://ap.wps.com/l/cbCainQYEPlEJPfG","pdf",578633,"English","# Inversion method for Gamma simulation\n# Cubic spline knots and bias–variance–error selection\n# LOOCV vs k-fold cross-validation statements\n# Regression spline coefficients from knots and degree","[{\"question\":\"How do you simulate a Gamma random variable using the inversion method in the given problem?\",\"answer\":\"Treat the Gamma(2, 1) variable as the sum of two independent exponential variables with rate 1, then use the inversion step −ln(U1) − ln(U2) with the provided uniform draws.\"},{\"question\":\"Which statistic corresponds to each line when selecting the number of knots for a cubic spline?\",\"answer\":\"Use the expected behavior: more knots increase flexibility and variance (line Y), while squared bias and training MSE decrease with flexibility (lines X and Z). Test MSE follows a U-shape (line W), and only the combination matching these conditions is correct.\"},{\"question\":\"What are the correct statements comparing LOOCV and k-fold cross-validation?\",\"answer\":\"LOOCV is k-fold cross-validation with k = n (true). For k \\u003c n, LOOCV does not have higher variance than k-fold (false). The validation set approach tends to overestimate test error more than LOOCV (false statement as written).\"}]","MAS I Sample Questions - Inversion Method, Spline Knots, Resampling, Regression Splines | PDF",1790489536]