[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572280-105":53,"doc-detail-572280-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","cs-355-topics-in-cryptography-probability-theory-cheat-sheet","CS 355: Topics in Cryptography - Probability-Theory Cheat Sheet","","Probability-theory cheat sheet for CS 355 (Spring 2020) outlining common notation and core tools for bounding random variables. Covers expectation linearity, the union bound, Markov, Chebyshev, and Chernoff inequalities, plus an applied example of distributing n balls into n bins. Uses indicator variables to compute the expected number and variance of empty bins, then derives concentration bounds and extends the approach to bound maximum bin load with tail estimates.",{"@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/cs-355-topics-in-cryptography-probability-theory-cheat-sheet/572280/",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/cs-355-topics-in-cryptography-probability-theory-cheat-sheet/572280.png","ImageObject",442,249,{"name":88,"@type":89},"jay","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What does the notation x RÃ S mean and when is it used?","Question",{"text":108,"@type":109},"It denotes choosing an element uniformly at random from a finite set S. It is used to define a random choice over the elements of S.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How does the linearity of expectation apply to constants and random variables?",{"text":113,"@type":109},"For random variables X and Y and constants a, b, it states E[aX + bY] = a·E[X] + b·E[Y]. This allows expectations to be computed term-by-term.",{"name":115,"@type":106,"acceptedAnswer":116},"How are Markov, Chebyshev, and Chernoff inequalities used to bound probabilities?",{"text":117,"@type":109},"Markov bounds the probability that a non-negative random variable exceeds a threshold. Chebyshev bounds deviations from the mean using variance. Chernoff provides exponential tail bounds for sums of independent variables bounded in [0,1], yielding strong concentration.","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},572280,1792403947,{"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":76,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":76,"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":9},3985747858866,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","CS 355: Topics in Cryptography Spring 2020  \nProbability-Theory Cheat Sheet  \nCommon Notation  \n• For a ﬁnite set S, x RÃ S denotes an element chosen uniformly at random from S. (More formally, x is a random variable taking each value a 2 S with probability 1/jSj.)  \n• For a positive integer n, the expression [ n ] denotes the set {1,2,..., n } .  \n• When we will want to be explicit about the probability distribution, we will often use either the notation E [x2 : x RÃ {¡1,1}] Æ 1 or Ex RÃ{¡1,1} [x2 ] Æ 1.  \n• For a probability event A, the indicator variable IA is a random variable that takes the value 1 when event A occurs, and 0 otherwise.  \nLinearity of expectation. let X and Y be random variables taking values in R, and let a, b 2 R be constants. Then  \nE [ aX Å bY ] Æ a ¢ E[X] Å bE [ Y ] .  \nUnion Bound. For events E1 ,   , En ,  \nPr[E1 [ ¢¢¢ [ En ] · Pr[E1 ] Å ¢¢¢ Å Pr[En ] .  \nIn many situation, we will be interested in bounding the probability that a random variable deviates from its expectation.  \nMarkov's inequality. Let X be a non-negative random variable and a È 0 be a constant. Then  \nE [X] Pr[X ¸ a] · .  \na  \nChebyshev's inequality. Let X be a random variable and let ± È 0 be a constant. Then  \nVar[X]  \nPr [jX ¡ E[X] j ¸ ±] ·  \n±2 .  \nChernoff Bound. Let X1 ,..., Xn be independent random variables taking values in [0, 1], and let X Æ Pi 2[n] Xi . Then  \n80 Ç ² · 1, Pr[X · (1 ¡ ²) E [X]] · e ¡ ²2~~ ~~E~~2[~~X~~]~~ and  \n²2 E[X]  \n80 Ç ², Pr[X ¸ (1 Å ²) E [X]] · e ¡ 2Å² .  \nNote that the Chernoff bound holds even if the Xi take real values in the interval [0, 1], rather than just integer values in {0, 1} . Moreover, the Xi don't have to be identically distributed, but they crucially need tobe mutually independent.  \nExample. Consider throwing n balls independently and uniformly into n bins.  \n• What is the expected number of empty bins?  \nFor i 2 [ n ], let Zi be and indicator variable of the event that bin i is empty. It holds  \nPr[Zi Æ 1] Æ ¡1 ¡  ¢n ¼  .  \n(More formally, we can use that for n È 1, j xj · n, it holds that ex ·¡1 Å xn¢n · ex ³1 ¡ xn2 ´) .  \nWe can express the number of empty bins Z as  \nZ Æ X Zi ,  \ni 2[n]  \nand, by linearity of expectation, the expected number of empty bins is:  \nE [Z] Æ E \"i X2[n] Zi \\# ¼ ne .  \n• How concentrated is the number of empty bins Z?  \nIdeally, we would want to show that Z is tightly concentrated around its expectation n / e.  \nThe ﬁrst attempt could be to use Markov's inequality. However, this only gives us a very weak bound, for example:  \nPr[Z ¸ 0.75n ] · Pr[Z ¸ ~~2~~ne] · E2n[Z/e] ¼ 1/2 .  \nNext, we will use Chebyshev's inequality to get a better bound. To this end, let's compute the variance of Z:  \nVar[Z] Æ E [Z2 ] ¡ (E [Z])2 .  \nTo compute the ﬁrst term,  \nE [Z2 ] Æ E \"i ,Xj 2[n] Zi ¢ Zj \\# Æ iXÆ6 j E [Zi Zj] Å i X2[n] E [Z2i] .  \nNow note that ZiZj Æ 1 when all n balls miss both bin i and j , which happens with probability  \n(1 ¡ 2/n)n . Since Zi 2 {0, 1}, it also holds that E [Z2i] Æ E [Zi] Æ (1 ¡ 1/ n)n . Therefore E [Z2 ] Æ n ( n ¡ 1)(1 ¡ )n Å n(1 ¡ )n , and E [Z]2 Æ n2 ¡1 ¡  ¢2n .  \nWith a little bit of work, one can then show that Var [Z] Æ O ( n) . Therefore, by Chebyshev's inequality,  \n  Var[Z]  \nPr[ ¯ Z ¡ ne ¯ ¸ ²n] · ²2 n2 Æ O(1/n) ,  \nwhich is a much tighter bound that we have obtained before from Markov's inequality.  \n• What it the maximum load amongst all bins?  \nWe will show that with high probability, no bin has load higher than O(ln n) (in fact, one can show a tighter bound of O(ln n / ln ln n)) .  \nFor i , j 2 [ n ] let Xi j be an indicator variable of ball i landing in bin j. Then, we can write the load of bin j as  \nLj Æ X Xi j ,  \ni 2[n]  \nand it holds  \nE [Lj] Æ X E [Xi j] Æ n ¢ Pr[Xi j Æ 1] Æ n ¢  Æ 1 .  \ni 2[n]  \nNow note that, for every j and every i Æ6 i 0 , the random variables Xi j and Xi0 j are independent random variables taking values in {0, 1}. Therefore, we can use the Chernoff bound, with E [Lj] Æ 1 and ² Æ 2 ln n Å 2, to clai","cbCain0nY7IVu3CG","https://ap.wps.com/l/cbCain0nY7IVu3CG","pdf",79884,"English","# Common Notation\n## Linearity of expectation\n## Union Bound\n# Probability Inequalities\n## Markov's inequality\n## Chebyshev's inequality\n## Chernoff Bound\n# Example: Balls into bins\n## Expected number of empty bins\n## Concentration of empty bins\n## Maximum load among bins","[{\"question\":\"What does the notation x RÃ S mean and when is it used?\",\"answer\":\"It denotes choosing an element uniformly at random from a finite set S. It is used to define a random choice over the elements of S.\"},{\"question\":\"How does the linearity of expectation apply to constants and random variables?\",\"answer\":\"For random variables X and Y and constants a, b, it states E[aX + bY] = a·E[X] + b·E[Y]. This allows expectations to be computed term-by-term.\"},{\"question\":\"How are Markov, Chebyshev, and Chernoff inequalities used to bound probabilities?\",\"answer\":\"Markov bounds the probability that a non-negative random variable exceeds a threshold. Chebyshev bounds deviations from the mean using variance. Chernoff provides exponential tail bounds for sums of independent variables bounded in [0,1], yielding strong concentration.\"}]","CS 355: Topics in Cryptography - Probability-Theory Cheat Sheet | PDF",1791926549]