[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572278-105":53,"doc-detail-572278-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","probability-cheat-sheet","Probability Cheat Sheet","","This Probability Cheat Sheet serves as a concise reference guide for fundamental statistical and probabilistic concepts. It covers essential formulas, including probability rules for complements, unions, and intersections, as well as expectation and variance properties for independent and dependent random variables. The document further details cumulative distribution functions, density functions, and the Central Limit Theorem. Designed for quick review, it provides a mathematical framework for analyzing random variables and their distributions.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/probability-cheat-sheet/572278/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/probability-cheat-sheet/572278.png","ImageObject",442,249,{"name":88,"@type":89},"Mafia Boss","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-23","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What is the variance of the sum of two variables X and Y when they are independent?","Question",{"text":108,"@type":109},"When variables X and Y are independent, the variance of their sum is the sum of their individual variances: Var(X + Y) = Var(X) + Var(Y).","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How is the probability of a complement event defined?",{"text":113,"@type":109},"The probability of a complement event Ac is defined as P(Ac) = 1 - P(A).",{"name":115,"@type":106,"acceptedAnswer":116},"What does the Central Limit Theorem state regarding independent variables?",{"text":117,"@type":109},"The Central Limit Theorem states that for a sequence of independent random variables with a defined mean and variance, the distribution of their sample mean approaches a normal distribution N(0, σ²) as the sample size n becomes large.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572278,1792447212,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},962090760730,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Probability Cheat Sheet  \n􀀏 P (Ac ) = 1 􀀀 P(A)  \n􀀏 P (A [ B) = P(A) + P(B) if A;B do not overlap.  \n􀀏 P (A \\ B) = P(A)P (B) if A;B are independent.  \n􀀏 E (aX + bY + c) = aEX + bEY + c  \n􀀏 Eg(X)  g (EX) unless g(􀀁) is a linear function  \n􀀏 Var (X) = E(X 􀀀 EX)2 = E(X)2 􀀀 (EX)2  \n􀀏 E (XY ) = (EX)(EY ) if X;Y are independent.  \n􀀏 Var (X + Y ) = Var(X) + Var(Y ) if X;Y are independent.  \n􀀏 Var (X + Y ) = Var(X) + Var(Y ) + 2cov(X;Y )  \n􀀏 Var (aX + c) = a2 Var (X)  \n􀀏 If X;Y independent =) g (X);h (Y ) also independent.  \n􀀏 CDF P (X 􀀔 t) = F(t) = FX (t)  \n􀀏 density f (t) = dFdt(t)  \n􀀏 EX =R xfX (x)dx =R xdF(x)  \n􀀏 Eg (X) =R g(x)dF (x)  \n􀀏 Central Limit Theorem: Suppose X1 ;X2 ;::: Xn ;::: are independent with a distribution F (x) . Let EX = 􀀖 and Var (X) = 􀀛 2 . If 0 \u003C 􀀛 2 \u003C 1 then (in terms of distribution) for large n  \n~~ ~~ 􀀖 ~~ ~~ 1  \npn(X 􀀀 􀀖) = pn[ n  \nn  \nXXi  \ni=1  \n􀀀 􀀖] 􀀙 N(0;􀀛2 )  \nPlus the table for \\Brand name distrbution/random variables\". (include all mean, variance, etc.)  \nPlus the map of how the di􀀋erent random variables are connected.  \n1","cbCaitymAGwNcYnJ","https://ap.wps.com/l/cbCaitymAGwNcYnJ","pdf",41881,"English","# Probability Foundations\n## Random Variable Properties\n# Distributions and Limits\n## Central Limit Theorem","[{\"question\":\"What is the variance of the sum of two variables X and Y when they are independent?\",\"answer\":\"When variables X and Y are independent, the variance of their sum is the sum of their individual variances: Var(X + Y) = Var(X) + Var(Y).\"},{\"question\":\"How is the probability of a complement event defined?\",\"answer\":\"The probability of a complement event Ac is defined as P(Ac) = 1 - P(A).\"},{\"question\":\"What does the Central Limit Theorem state regarding independent variables?\",\"answer\":\"The Central Limit Theorem states that for a sequence of independent random variables with a defined mean and variance, the distribution of their sample mean approaches a normal distribution N(0, σ²) as the sample size n becomes large.\"}]","Probability Cheat Sheet | PDF",1791926541]