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The notes state that sample averages concentrate around the underlying mean with high probability, and that sums or averages become approximately normal as sample size grows. An example with Bernoulli trials illustrates how probabilities of being within a tolerance increase with n. Emphasizes systematic error versus random error and provides guidance for computation.",{"@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/central-limit-theorem-and-the-law-of-large-numbers-class-6/268530/",{"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/central-limit-theorem-and-the-law-of-large-numbers-class-6/268530.png","ImageObject",300,407,{"name":92,"@type":93},"\tCallum ","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the main learning goals for this class?","Question",{"text":112,"@type":113},"Students learn the statements of the law of large numbers and the central limit theorem, and how to use the CLT to approximate probabilities involving averages and sums of i.i.d. random variables.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does the law of large numbers say about the sample average Xn?",{"text":117,"@type":113},"As n increases, the probability that Xn is close to the true mean μ approaches 1, meaning the sample average concentrates near μ with high probability.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the notes’ Bernoulli example illustrate LoLN?",{"text":121,"@type":113},"By computing probabilities that the head proportion Xn lies within fixed tolerances (e.g., 0.1 or 0.01) of 0.5, the example shows these probabilities increase toward 1 as n grows.","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},268530,1790510682,{"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":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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":41},137451211410,"https://ap-avatar.wpscdn.com/avatar/2000bb0a9246f588df?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786362646172706240","Central Limit Theorem and the Law of Large Numbers  \nClass 6, 18.05  \nJeremy Orlo􀀋 and Jonathan Bloom  \n1 Learning Goals  \n1. Understand the statement of the law of large numbers.  \n2. Understand the statement of the central limit theorem.  \n3. Be able to use the central limit theorem to approximate probabilities of averages and sums of independent identically-distributed random variables.  \n2 Introduction  \nWe all understand intuitively that the average of many measurements of the same unknown quantity tends to give a better estimate than a single measurement. Intuitively, this is because the random error of each measurement cancels out in the average. In these notes we will make this intuition precise in two ways: the law of large numbers (LoLN) and the central limit theorem (CLT) .  \nBrie􀀍y, both the law of large numbers and central limit theorem are about many independent samples from same distribution. The LoLN tells us two things:  \n1. The average of many independent samples is (with high probability) close to the mean of the underlying distribution.  \n2. This density histogram of many independent samples is (with high probability) close to the graph of the density of the underlying distribution.  \nTo be absolutely correct mathematically we need to make these statements more precise, but as stated they are a good way to think about the law of large numbers.  \nThe central limit theorem says that the sum or average of many independent copies of a random variable is approximately a normal random variable. The CLT goes on to give precise values for the mean and standard deviation of the normal variable.  \nThese are both remarkable facts. Perhaps just as remarkable is the fact that often in practice n does not have to be all that large.  \n2.1 There is more to experimentation than mathematics  \nThe mathematics of the LoLN says that the average of a lot of independent samples from a random variable will almost certainly approach the mean of the variable. The mathematics cannot tell us if the tool or experiment is producing data worth averaging. For example, if the measuring device is defective or poorly calibrated then the average of many measurements will be a highly accurate estimate of the wrong thing! This is an example of  \n18.05 class 6, Central Limit Theorem and the Law of Large Numbers, Spring 2017 2  \nsystematic error or sampling bias, as opposed to the random error controlled by the law of large numbers.  \n3 The law of large numbers  \nSuppose X1 , X2 , . . . , Xn are independent random variables with the same underlying distribution. In this case, we say that the Xi are independent and identically-distributed, or i.i.d. In particular, the Xi all have the same mean 􀀖 and standard deviation 􀀛 .  \nLet Xn be the average of X1 ; : : : ; Xn:  \nXn = X1~~ ~~+~~ ~~X2~~ ~~+n~~ ~~􀀁~~ ~~􀀁~~ ~~􀀁~~ ~~+~~ ~~Xn = 1n Xi1 Xi:  \nNote that X n is itself a random variable. The law of large numbers and central limit theorem tell us about the value and distribution of Xn, respectively.  \nLoLN: As n grows, the probability that X n is close to 􀀖 goes to 1 .  \nCLT: As n grows, the distribution of Xn converges to the normal distribution N(􀀖; 􀀛2 =n) . Before giving a more formal statement of the LoLN, let's unpack its meaning through a concrete example (we'll return to the CLT later on) .  \nExample 1 . Averages of Bernoulli random variables  \nSuppose each Xi is an independent 􀀍ip of a fair coin, so Xi 􀀘 Bernoulli(0:5) and 􀀖 = 0:5. Then X n is the proportion of heads in n 􀀍ips, and we expect that this proportion is close to 0.5 for large n. Randomness being what it is, this is not guaranteed; for example we could get 1000 heads in 1000 􀀍ips, though the probability of this occurring is very small.  \nSo our intuition translates to: with high probability the sample average X n is close to the mean 0:5 for large n. We'll demonstrate by doing some calculations in R. You can 􀀌nd the code used for `class 6 prep' in the usual place on our MITx site.  \nTo","cbCaigAGk9xYmEJb","https://ap.wps.com/l/cbCaigAGk9xYmEJb","pdf",358025,12,"English","# Learning Goals\n# Introduction\n# The law of large numbers\n## Averages of Bernoulli random variables\n## Formal statement of the law of large numbers","[{\"question\":\"What are the main learning goals for this class?\",\"answer\":\"Students learn the statements of the law of large numbers and the central limit theorem, and how to use the CLT to approximate probabilities involving averages and sums of i.i.d. random variables.\"},{\"question\":\"What does the law of large numbers say about the sample average Xn?\",\"answer\":\"As n increases, the probability that Xn is close to the true mean μ approaches 1, meaning the sample average concentrates near μ with high probability.\"},{\"question\":\"How does the notes’ Bernoulli example illustrate LoLN?\",\"answer\":\"By computing probabilities that the head proportion Xn lies within fixed tolerances (e.g., 0.1 or 0.01) of 0.5, the example shows these probabilities increase toward 1 as n grows.\"}]","Central Limit Theorem and the Law of Large Numbers - Class 6 | PDF",1789432046]