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The material introduces random experiments, sample spaces, power sets, sigma-algebras (including Borel sigma-algebras), event spaces, and probability measures/spaces. It further covers probability mass functions, random variables and vectors, distribution functions, and the foundational notation used throughout related chapters.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/stochastics-summer-term-2019-cheat-sheets-tuhh-class-27th-august-2019/572937/",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/stochastics-summer-term-2019-cheat-sheets-tuhh-class-27th-august-2019/572937.png","ImageObject",442,249,{"name":88,"@type":89},"Dipper","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","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 does the cheat sheet define as a random experiment and a sample space?","Question",{"text":108,"@type":109},"A random experiment is an experimental procedure with a definite set of possible outcomes but an unpredictable single outcome. The sample space is the set of possible outcomes.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What is a sigma-algebra and how is the Borel sigma-algebra defined?",{"text":113,"@type":109},"A sigma-algebra on a sample space is a collection of subsets closed under complement and countable unions. The Borel sigma-algebra on R^n is generated by a specified collection of rational interval products and is used to define measurable sets.",{"name":115,"@type":106,"acceptedAnswer":116},"When is a function X called a random variable or random vector?",{"text":117,"@type":109},"X is a random variable if preimages of Borel sets in R are measurable events. It is a random vector (or random variable) if preimages of Borel sets in R^n are measurable under the event space sigma-algebra.","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},572937,1792241886,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"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":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":141},1374404997633,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Stochastics Summer Term 2019 Cheat Sheets  \nTUHH Class  \n27th August 2019 3:18pm GMT  \nContents  \n1 Thesaurus / De􀀌nitions 2  \n1.1 Probabillity Theory ......................................... 2  \n1.2 Statistics ............................................... 5  \n1.2.1 Point Estimators ....................................... 6  \n1.2.2 Interval Estimators ...................................... 7  \n1.2.3 Statistical Hypothesis Testing ................................ 7  \n2 Formulas / Theorems 8  \n2.1 Stochastic ............................................... 8  \n2.1.1 Convergence of Random Variables ............................. 11  \n2.2 Math .................................................. 11  \n3 Distributions 12  \n3.1 Discrete Uniform Distribution .................................... 12  \n3.2 Bernoulli Distribution ........................................ 13  \n3.3 Multinomial Distribution ...................................... 13  \n3.4 Binomial Distribution ........................................ 13  \n3.5 Hypergeometric Distribution .................................... 13  \n3.6 Poisson Distribution ......................................... 14  \n3.7 Continuous Uniform Distribution .................................. 14  \n3.8 Waiting Times ............................................ 14  \n3.8.1 Negative Binomial Distribution ............................... 14  \n3.8.2 Geometric Distribution ................................... 15  \n3.8.3 Gamma Distribution ..................................... 15  \n3.8.4 Exponential Distribution .................................. 15  \n3.9 Normal Distribution ......................................... 16  \n3.10 Multidimensional Normal Distribution ............................... 16  \n3.11 Distribution Properties ........................................ 16  \n3.11.1 Important CDFs ....................................... 16  \n3.11.2 Convolution of RVs (X+Y) ................................. 16  \n3.11.3 Special Cases of Distributions ................................ 16  \n3.11.4 Inequalities .......................................... 17  \n4 Other Probabilities 17  \n4.1 Conditional Probabilities ....................................... 17  \n4.2 Independence ............................................. 17  \n5 R 17  \n5.1 General Advice ............................................ 17  \n5.2 Distributions ............................................. 18  \n5.3 Graphs ................................................. 18  \n5.4 Stochastics Functions ......................................... 18  \n5.5 Math .................................................. 18  \n5.6 Operators ............................................... 18  \n5.7 Other useful functions ........................................ 19  \n6 Calculus and Basic Functions 19  \n1 Thesaurus / De􀀌nitions  \n1.1 Probabillity Theory  \nRandom Experiment: A random experiment is an experimental procedure with a de􀀌nite set of possible outcomes but an unpredictable single outcome.  \nsample space 􀀊: The set of possible outcomes of a random experiment. In general, any 􀀊  ; is a sample space.  \nsample: A (random) sample of length n is a list of single outcomes of n realizations of the experiment:  \n(x1 ; : : : ; xn); with x 1 ; : : : ; xn 2 􀀊 . Outcomes can occur multiple times.  \nPower set: The set of all subsets of any set 􀀊, including the empty set and 􀀊 itself, is called the power set, also denoted as P(􀀊) .  \n􀀛-algebra (􀀛-􀀌eld): Let 􀀊 be a sample space. Then A 􀀒 P(􀀊) is a 􀀛-algebra on 􀀊 if:  \n1. ; ; 􀀊 2 A,  \n2. A 2 A =) 􀀊 n A 2 A,  \n3. (An) 2 A =) SAn 2 A.  \nFor some G 􀀒 P(􀀊), the unique smallest 􀀛-algebra A containing G is called the 􀀛-algebra generated by G. It is denoted by 􀀛 (G) := A.  \nIn the discrete case, the 􀀛-algebra is given by A := P(􀀊) .  \nBorel 􀀛-algebra on Rn: Let 􀀛 = Rn ; G = f Qnj=1 (aj ; bj ]; aj ; bj 2 Q; aj \u003C bj (j 2 f1; : : : ; ng)g. Then the Borel 􀀛-algebra B(Rn ) on Rn is de􀀌ned as B (Rn ) := 􀀛(G) . The Borel 􀀛-algebra on subsets 􀀊 􀀒 Rn is de􀀌","cbCaihvrCf6yQ60N","https://ap.wps.com/l/cbCaihvrCf6yQ60N","pdf",371699,20,"English","# Contents\n## Thesaurus / Definitions\n## Formulas / Theorems\n## Distributions\n## Other Probabilities\n## R\n## Calculus and Basic Functions","[{\"question\":\"What does the cheat sheet define as a random experiment and a sample space?\",\"answer\":\"A random experiment is an experimental procedure with a definite set of possible outcomes but an unpredictable single outcome. The sample space is the set of possible outcomes.\"},{\"question\":\"What is a sigma-algebra and how is the Borel sigma-algebra defined?\",\"answer\":\"A sigma-algebra on a sample space is a collection of subsets closed under complement and countable unions. The Borel sigma-algebra on R^n is generated by a specified collection of rational interval products and is used to define measurable sets.\"},{\"question\":\"When is a function X called a random variable or random vector?\",\"answer\":\"X is a random variable if preimages of Borel sets in R are measurable events. It is a random vector (or random variable) if preimages of Borel sets in R^n are measurable under the event space sigma-algebra.\"}]","Stochastics Summer Term 2019 Cheat Sheets - TUHH Class - 27th August 2019 | PDF",1791930166,7]