{"id":77538,"date":"2021-02-25T14:45:08","date_gmt":"2021-02-25T19:45:08","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=77538"},"modified":"2023-02-24T14:44:33","modified_gmt":"2023-02-24T19:44:33","slug":"central-limit-theorem-explained-in-python-with-examples-part-ii","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples-part-ii\/","title":{"rendered":"Central Limit Theorem explained in Python (with examples) &#8211; Part II"},"content":{"rendered":"\n<p><em>See <a href=\"\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples\/\">Part I<\/a> to get started. <\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"central-limit-theorem-statement-assumptions\">Central Limit Theorem: Statement &amp; Assumptions<\/h2>\n\n\n\n<p>Suppose we are taking repeated samples of size &#8216;n&#8217; from a population with any kind of probability distribution. Then, the Central Limit Theorem states that given a high enough sample size, the following properties hold true:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sampling distribution&#8217;s mean = Population mean&nbsp;(\u03bc)(\u03bc), and<\/li>\n\n\n\n<li>Sampling distribution&#8217;s standard deviation (standard error) =&nbsp;\u03c3\/\u221an\u03c3\/\u221an, such that for n \u2265 30, the sampling distribution tends to a normal distribution for all practical purposes.<\/li>\n<\/ul>\n\n\n\n<p>In the next section, we will try to understand the workings of the CLT with the help of simulations in Python.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"demonstration-of-clt-in-action-using-simulations-in-python-with-examples\">Demonstration of CLT in action using simulations in Python with examples<\/h2>\n\n\n\n<p>The main point demonstrated in this section will be that for a population following any distribution, the sampling distribution (sample mean&#8217;s distribution) will tend to be normally distributed for a large enough sample size.<\/p>\n\n\n\n<p>We will consider two examples and check whether the CLT holds.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"example-1-exponentially-distributed-population\">Example 1: Exponentially distributed population<\/h3>\n\n\n\n<p>Suppose we are dealing with a population which is exponentially distributed. Exponential distribution is a continuous distribution that is often used to model the expected time one needs to wait before the occurrence of an event.<\/p>\n\n\n\n<p>The main parameter of exponential distribution is the &#8216;rate&#8217; parameter&nbsp;\u03bb\u03bb, such that both the mean and the standard deviation of the distribution are given by&nbsp;(1\/\u03bb)(1\/\u03bb).<\/p>\n\n\n\n<p>The following represents our exponentially distributed population:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"370\" height=\"228\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-1-1.png\" alt=\"\" class=\"wp-image-185578 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-1-1.png 370w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-1-1-300x185.png 300w\" data-sizes=\"(max-width: 370px) 100vw, 370px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 370px; aspect-ratio: 370\/228;\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"823\" height=\"165\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-2.png\" alt=\"\" class=\"wp-image-185579 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-2.png 823w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-2-700x140.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-2-300x60.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/population-distribution-quantInsti-2-768x154.png 768w\" data-sizes=\"(max-width: 823px) 100vw, 823px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 823px; aspect-ratio: 823\/165;\" \/><\/figure>\n\n\n\n<p>We can see that the distribution of our population is far from normal! In the following code, assuming that&nbsp;\u03bb\u03bb=0.25, we calculate the mean and the standard deviation of the population:<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">\t\n# Importing necessary libraries<br>\nimport warnings<br>\nwarnings.filterwarnings(&#8216;ignore&#8217;)<br>\nimport pandas as pd, numpy as np<br>\nimport matplotlib.pyplot as plt<br>\n%matplotlib inline<br>\nimport seaborn as sns<br><br>\n\n\n# rate parameter for the exponentially distributed population<br>\nrate = 0.25<br><br>\n\n#Population mean<br>\nmu = 1\/rate<br><br>\n\n# Population standard deviation<br>\nsd = np.sqrt(1\/(rate**2))<br><br>\n\nprint(&#8216;Population mean:&#8217;, mu)<br>\nprint(&#8216;Population standard deviation:&#8217;, sd)\n<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">\t\nPopulation mean: 4.0<br>\nPopulation standard deviation: 4.0\n<\/p>\n\n\n\n<p><em>Stay tuned for the next installment, in which Ashutosh Dave will see how the sampling distribution looks for this population and will consider two cases, i.e. with a small sample size (n= 2), and a large sample size (n=500).<\/em><\/p>\n\n\n\n<p><em>Visit QuantInsti for additional details and to download full code:&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/central-limit-theorem\/\">https:\/\/blog.quantinsti.com\/central-limit-theorem\/<\/a><\/em>.<\/p>\n\n\n\n<p><em><small>Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.<\/small><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ashutosh Dave from QuantInsti demonstrates CLT in action using simulations in Python. Ready-to-use Python code is available for download.<\/p>\n","protected":false},"author":295,"featured_media":36044,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[343,349,338,350,341,344],"tags":[231,9171,806,595,9172],"contributors-categories":[13654],"class_list":{"0":"post-77538","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-programing-languages","8":"category-python-development","9":"category-ibkr-quant-news","10":"category-quant-asia-pacific","11":"category-quant-development","12":"category-quant-regions","13":"tag-analysis","14":"tag-central-limit-theorem","15":"tag-data-science","16":"tag-python","17":"tag-sampling-distribution","18":"contributors-categories-quantinsti"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.5) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Central Limit Theorem explained in Python (with examples) &#8211; Part II<\/title>\n<meta name=\"description\" content=\"Ashutosh Dave from QuantInsti demonstrates CLT in action using simulations in Python. 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