[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-398740-105":59,"doc-detail-398740-en":131},{"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":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","exam-p-formula-review-sheet-updated-06282023","Exam P Formula & Review Sheet - updated 06/28/2023","","Reference sheet compiling key probability formulas and definitions for Exam P. Covers conditional probability, Bayes’ Rule, binomial coefficients, DeMorgan’s laws, inclusion–exclusion, and the central limit theorem. Summarizes univariate random variables including distribution and survival functions, expectation, moments, median and mode concepts, and a broad set of discrete and continuous distributions (binomial, geometric, Poisson, negative binomial, hypergeometric, uniform, normal, exponential, gamma, lognormal, beta). Includes selected insurance risk concepts like deductible, policy limit, coinsurance, and cost expectations.",{"@graph":69,"@context":123},[70,84,106],{"@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/exam-p-formula-review-sheet-updated-06282023/398740/",{"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/exam-p-formula-review-sheet-updated-06282023/398740.png","ImageObject",300,407,{"name":92,"@type":93},"lilac","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-07","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",18,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What topics from general probability are covered in the Exam P formula sheet?","Question",{"text":113,"@type":114},"It includes conditional probability, Bayes’ rule, binomial coefficients, DeMorgan’s laws, inclusion–exclusion equations, and the central limit theorem.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Which univariate random variable concepts and measures are summarized?",{"text":118,"@type":114},"The sheet covers distribution and survival functions, expected value, moments, median and mode definitions, and links between mean/variance and key theorems.",{"name":120,"@type":111,"acceptedAnswer":121},"Does the sheet include both discrete and continuous probability distributions?",{"text":122,"@type":114},"Yes. It lists common discrete distributions (e.g., binomial, geometric, Poisson, negative binomial, hypergeometric) and continuous distributions (e.g., uniform, normal, exponential, gamma, lognormal, beta), along with relevant parameter interpretations.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},398740,1790835060,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":52,"preview_page_count":4},3985747861865,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Formula & Review Sheet for Exam P  \nYour quick-reference guide for the most important formulas and concepts you need.  \nWant More Than Just the Formula Sheet?  \nThis free Formula & Review Sheet is just one part of the ACTEX Learning experience. Our complete study program is designed to help you  \nunderstand concepts, practice effectively, and pass with confidence. With the ACTEX Learning Bundle for Exam P, you get:  \n Comprehensive Study Manual  \n GOAL Practice Problems  \n Instructional Videos  \n Flashcards  \n Formula & Review Sheets  \n Learning Platform with a Study Planner and Topic Explorer  \nWant an Extra Edge?  \nTake your preparation even further with premium ACTEX resources designed to sharpen exam skills and boost confidence when it matters most.  \nPopular Upgrades:  \n Live Bootcamps  \nLearn More  \nOne-On-One Tutoring  \n Graded Mock Exams  \nClick anywhere on this page or scan the QR code to learn more about offerings for this exam.  \nExam P Formula & Review Sheet  \n(updated 06/28/2023)  \nGENERAL PROBABILITY  \n\n| Conditional Probability of Event B Given Event A: P [BjA] = P~~ ~~[BP[\\A]A] Probability of Event B Partitioned by Events A and A0 :\u003Cbr>P [B] = P [B \\ A] + P [B \\ A0] = P [BjA] 􀀁 P [A] + P [BjA0] 􀀁 P [A0]\u003Cbr>P [A \\ B]  P [Bj A] 􀀁 P [A] \u003Cbr>Bayes’Rule: P [Aj B] = P [B] = P [Bj A] 􀀁 P [A] + P [Bj A0] 􀀁 P [A0]\u003Cbr>Binomial Coefficient-Number of Subsets of Size k From a Collection of n Objects: 􀀒nk􀀓 = k!􀀁~~ ~~(nn~~ ~~􀀀!~~ ~~k)! DeMorgan’s Laws: (A [ B)0 = A0 \\ B 0 and (A \\ B)0 = A0 [ B 0 .\u003Cbr>Inclusion-Exclusion Equations: P (A [ B) = P(A) + P(B) 􀀀 P(A \\ B)\u003Cbr>P [A [ B [ C] = P[A] + P[B] + P[C] 􀀀 P[A \\ B] 􀀀 P[B \\ C] 􀀀 P[A \\ C] + P[A \\ B \\ C]\u003Cbr>Central Limit Theorem: Suppose that X is a random variable with mean 􀀖 and standard deviation 􀀛 and\u003Cbr>suppose that X1 ; X2 ; 􀀁 􀀁 􀀁 ; Xn are n independent random variables with the same dis\u003Cbr>tribution as X .\u003Cbr>Let Yn = X1 + X2 + 􀀁 􀀁 􀀁 + Xn.\u003Cbr>Then E [Yn] = n􀀖 and Var [Yn] = n􀀛 2 , and as n increases, the distribution of Yn approaches a normal distribution N(n􀀖; n􀀛2 ) .\u003Cbr>\u003Cbr>UNIVARIATE RANDOM VARIABLES |  |  |  |  |\n| --- | --- | --- | --- | --- |\n| Distribution Function & Survival Function of Random Variable X:\u003Cbr>FX (t) = P [X 􀀔 t]\u003Cbr>Expected Value of Random Variable X:\u003Cbr>E [X] = P xi 􀀁 pX (xi) (discrete)\u003Cbr>all xi |  |  |  | SX (t) = 1 􀀀 FX (t) = P [X > t]\u003Cbr>E [X] =R1􀀀1 x􀀁 fX (x) dx (continuous) |\n| Moment of Random Variable X:\u003Cbr>Coefficient of Variation of X : | If n 􀀕 1 is an integer, the n-th moment of X is E [Xn ]:\u003Cbr>If the mean of X is 􀀖, then the n-th central moment of X is E[(X 􀀀 􀀖)n ]: |  |  |  |\n|  | 􀀛X 􀀖X | or | 􀀛X\u003Cbr>E [X] |  |\n| Median of the Distribution of Random Variable X: Smallest m for which FX (m) = :5 |  |  |  |  |\n\nNeed Help? Email [support@actuarialuniversity.com](support@actuarialuniversity.com) Copyright © ArchiMedia Advantage Inc. All Rights Reserved  \nACTEX Learning  \nExam P  \nPage 2  \nMode of the Distribution of Random Variable X: The point c at which pX (c) (discrete)  \nor fX (c) (continuous) is maximized.  \nDiscrete Uniform Distribution on the Integers 1 ; 2; : : : ; N:  \n1   N + 1  \npX (k) = for k = 1 ; 2; : : : N E [X] =  \nN 2  \nBinomial Distribution for n Trials and Success Probability p:  \npX (k) = 􀀒nk􀀓 pk (1 􀀀 p)n􀀀k E [X] = np Geometric Distribution on the Integers 0 ; 1 ; 2 ; : : : and Parameter p:  \npX (k) = (1 􀀀 p)k 􀀁 p E [X] = 1~~ ~~~~ ~~p  \nPoisson Distribution with Mean 􀀕 :  \npX (k) = e~~ ~~􀀀􀀕k!􀀕~~ ~~k for k = 0 ; 1 ; 2 ; : : : E [X] = 􀀕  \nNegative Binomial Distribution with Parameters p and integer 􀀍 􀀕 1 : pX (x) = 􀀒􀀍 + xx 􀀀 1􀀓 p􀀍 (1 􀀀 p)x for x = 0 ; 1 ; 2 ; 3 ; 􀀁 􀀁 􀀁 E [X] = 􀀍(1~~ ~~p)  \nHypergeometric Distribution:  \npX (x) = 􀀀 Kx􀀁􀀀MnM􀀀Kx􀀁  E [X] = nK  \n􀀀 n 􀀁 M for maxf0; n 􀀀 (M 􀀀 K)g 􀀔 x 􀀔 minfn; Kg  \nContinuous Uniform Distribution on the Interval [a; b]:  \n 1   a + b  \nfX (x) = for a 􀀔 x 􀀔 b E [X] =  \nb 􀀀 a 2  \nStandard Normal Distribution N(0; 1) (Mean 0, Variance 1):  \n􀀞 (x) = p12􀀙 e 􀀀x2 /2 for 􀀀1 \u003C x \u003C 1 FX (x) = 􀀈(","cbCaic1KEf8DqTuP","https://ap.wps.com/l/cbCaic1KEf8DqTuP","pdf",1302542,"English","# GENERAL PROBABILITY\n## Conditional probability and Bayes' rule\n## Inclusion-exclusion, DeMorgan's laws, binomial coefficients\n## Central limit theorem\n# UNIVARIATE RANDOM VARIABLES\n## Distribution/survival functions, expectation and moments\n## Median and mode concepts\n## Discrete distributions\n## Continuous distributions\n# MULTIVARIATE RANDOM VARIABLES\n## Marginal distributions","[{\"question\":\"What topics from general probability are covered in the Exam P formula sheet?\",\"answer\":\"It includes conditional probability, Bayes’ rule, binomial coefficients, DeMorgan’s laws, inclusion–exclusion equations, and the central limit theorem.\"},{\"question\":\"Which univariate random variable concepts and measures are summarized?\",\"answer\":\"The sheet covers distribution and survival functions, expected value, moments, median and mode definitions, and links between mean/variance and key theorems.\"},{\"question\":\"Does the sheet include both discrete and continuous probability distributions?\",\"answer\":\"Yes. It lists common discrete distributions (e.g., binomial, geometric, Poisson, negative binomial, hypergeometric) and continuous distributions (e.g., uniform, normal, exponential, gamma, lognormal, beta), along with relevant parameter interpretations.\"}]","Exam P Formula & Review Sheet - updated 06/28/2023 | PDF",1790488591]