{"id":79038,"date":"2021-03-09T10:55:47","date_gmt":"2021-03-09T15:55:47","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=79038"},"modified":"2023-02-24T14:40:37","modified_gmt":"2023-02-24T19:40:37","slug":"central-limit-theorem-explained-in-python-with-examples-part-iii","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples-part-iii\/","title":{"rendered":"Central Limit Theorem explained in Python (with examples) \u2013 Part III"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>See&nbsp;<a href=\"\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples\/\">Part I<\/a>&nbsp;and <a href=\"\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples-part-ii\/\">Part II<\/a> to get started.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now we want to see how the sampling distribution looks for this population. We will consider two cases, i.e. with a small sample size (n= 2), and a large sample size (n=500).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First, we will draw 50 random samples from our population of size 2 each. The code to do the same in Python is given below:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">\t\n# drawing 50 random samples of size 2 from the exponentially distributed population<br>\nsample_size = 2<br>\ndf2 = pd.DataFrame(index= [&#8216;x1&#8217;, &#8216;x2&#8217;] )<br><br>\nfor i in range(1, 51):<br>\n    exponential_sample = np.random.exponential((1\/rate), sample_size)<br>\n    col = f&#8217;sample {i}&#8217;<br>\n    df2[col] = exponential_sample<br><br>\n# Taking a peek at the samples<br>\ndf2\n<\/p><\/p>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\"><\/div><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th><\/th><th>sample 1<\/th><th>sample 2<\/th><th>sample 3<\/th><th>sample 4<\/th><th>sample 5<\/th><th>sample 6<\/th><th>sample 7<\/th><th>sample 8<\/th><th>sample 9<\/th><th>sample 10<\/th><th>&#8230;<\/th><th>sample 41<\/th><th>sample 42<\/th><th>sample 43<\/th><th>sample 44<\/th><th>sample 45<\/th><th>sample 46<\/th><th>sample 47<\/th><th>sample 48<\/th><th>sample 49<\/th><th>sample 50<\/th><\/tr><\/thead><tbody><tr><th>x1<\/th><td>3.308423<\/td><td>7.105807<\/td><td>0.787859<\/td><td>2.811602<\/td><td>0.255161<\/td><td>5.085278<\/td><td>7.253975<\/td><td>2.549191<\/td><td>1.318133<\/td><td>0.659430<\/td><td>&#8230;<\/td><td>13.017465<\/td><td>10.280906<\/td><td>1.863208<\/td><td>4.000935<\/td><td>1.119582<\/td><td>1.640825<\/td><td>7.242127<\/td><td>0.807044<\/td><td>11.797688<\/td><td>4.585229<\/td><\/tr><tr><th>x2<\/th><td>2.969489<\/td><td>1.082994<\/td><td>3.382971<\/td><td>3.474494<\/td><td>8.949835<\/td><td>0.993594<\/td><td>7.335135<\/td><td>5.529222<\/td><td>3.760836<\/td><td>1.690919<\/td><td>&#8230;<\/td><td>8.690013<\/td><td>1.468530<\/td><td>0.376954<\/td><td>0.167118<\/td><td>4.100110<\/td><td>0.255927<\/td><td>1.754906<\/td><td>3.647159<\/td><td>1.883523<\/td><td>1.101046<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">2 rows \u00d7 50 columns<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For each of the 50 samples, we can calculate the sample mean and plot its distribution as follows:<\/p>\n\n\n\n<p><p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">\t\n# Calculating sample means and plotting their distribution<br>\ndf2_sample_means = df2.mean()<br>\nsns.distplot(df2_sample_means);\n<\/p><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"378\" height=\"252\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/03\/sample-distribution-clt-quantinsti.png\" alt=\"\" class=\"wp-image-79087 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/03\/sample-distribution-clt-quantinsti.png 378w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/03\/sample-distribution-clt-quantinsti-300x200.png 300w\" data-sizes=\"(max-width: 378px) 100vw, 378px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 378px; aspect-ratio: 378\/252;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">We can observe that even for a small sample size such as 2, the distribution of sample means looks very different from that of the exponential population, and looks more like a poor approximation of a normal distribution, with some positive skew.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Stay tuned for the next installment, in which Ashutosh will repeat the above process, but with a much larger sample size (n=500).<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><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 class=\"wp-block-paragraph\"><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 showcases how to do exponential sampling in Python and offers ready-to-use code.<\/p>\n","protected":false},"author":295,"featured_media":25709,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[343,349,338,350,341,344],"tags":[806,4941,865,595],"contributors-categories":[13654],"class_list":["post-79038","post","type-post","status-publish","format-standard","has-post-thumbnail","category-programing-languages","category-python-development","category-ibkr-quant-news","category-quant-asia-pacific","category-quant-development","category-quant-regions","tag-data-science","tag-financial-mathematics","tag-github","tag-python","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 v28.0) - 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