2D Normal Distributions

⏭️ 𝘾𝙤𝙣𝙣𝙚𝙘𝙩 𝙬𝙞𝙩𝙝 𝙪𝙨 𝙤𝙣 𝙋𝘼𝙏𝙍𝙀𝙊𝙉   / socratica   We've discussed the one-dimensional Normal Distribution (the bell curve) in a previous video, so you're all experts now! Normal Distributions    • Normal Distributions Explained – With Real...   In this lesson, we extend the familiar Bell Curve into two dimensions. The 2D normal distribution — also called the Gaussian distribution in two variables — describes how data spreads when there’s variation in two directions at once. Starting with darts on a board and lengths of wood, we build up the intuition for moving from 1D to 2D. You’ll learn how the mean vector and covariance matrix define the shape of the distribution, and how probability density functions generalize from curves to surfaces. Along the way, we explore real-world applications, including stock market returns and precision sports. By the end, you’ll see how the 2D case prepares us for the general multivariate normal distribution in any number of dimensions ⏭️ 𝙔𝙤𝙪 𝙘𝙖𝙣 𝙟𝙪𝙢𝙥 𝙩𝙤 𝙨𝙚𝙘𝙩𝙞𝙤𝙣𝙨 𝙤𝙛 𝙩𝙝𝙚 𝙫𝙞𝙙𝙚𝙤 𝙝𝙚𝙧𝙚: 0:00 Darts and 2D variation 1:00 Recap of 1D normal distributions 2:00 From curves to surfaces 3:00 The mean vector 4:30 The covariance matrix 6:00 Stock returns example 7:30 The 2D probability density function 9:00 Applications: darts & stocks 11:00 From 2D to N-dimensional distributions ▶️ 𝙒𝘼𝙏𝘾𝙃 𝙉𝙀𝙓𝙏: Normal Distributions    • Normal Distributions Explained – With Real...   Special thanks to our wonderful Patreon supporters: Umar Khan Tracy Karin Prell Thomas Myers Michael Shebanow Marcos Silveira M Andrews KW Kevin B John Krawiec Jeremy Shimanek Eric Eccleston Christopher Kemsley Jim Woodworth Thank you, kind friends! 💜🦉 𝘽𝙚𝙘𝙤𝙢𝙚 𝙤𝙪𝙧 𝙋𝙖𝙩𝙧𝙤𝙣 𝙤𝙣 𝙋𝙖𝙩𝙧𝙚𝙤𝙣:   / socratica   📚 𝙒𝙚 𝙧𝙚𝙘𝙤𝙢𝙢𝙚𝙣𝙙 (affiliate links): The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow https://amzn.to/4j9n0YP The Art of Statistics: How to Learn from Data by David Spiegelhalter https://amzn.to/3S9E46a How to Be a Great Student (from Socratica!) ebook: https://amzn.to/2Lh3XSP paperback: https://amzn.to/3t5jeH3 🎬 𝘾𝙍𝙀𝘿𝙄𝙏𝙎: Written & Produced by: Michael Harrison & Kimberly Hatch Harrison Edited by: Alivia Brown and Megi Shuke Music License from Soundstripe Code: 6F6NQWRP2DBJBIQZ 🎓 𝘼𝘽𝙊𝙐𝙏 𝙊𝙐𝙍 𝙄𝙉𝙎𝙏𝙍𝙐𝘾𝙏𝙊𝙍𝙎: Michael earned his BS in Math from Caltech, and did his graduate work in Math at UC Berkeley and University of Washington, specializing in Number Theory. A self-taught programmer, Michael taught both Math and Computer Programming at the college level. He applied this knowledge as a financial analyst (quant) and as a programmer at Google. Kimberly earned her BS in Biology and another BS in English at Caltech. She did her graduate work in Molecular Biology at Princeton, specializing in Immunology and Neurobiology. Kimberly spent 16+ years as a research scientist and a dozen years as a biology and chemistry instructor. Michael and Kimberly Harrison co-founded Socratica. Their mission? To create the education of the future. Ready to 🧠 𝙇𝙀𝘼𝙍𝙉 𝙈𝙊𝙍𝙀 with Socratica? 📺 𝙎𝙪𝙗𝙨𝙘𝙧𝙞𝙗𝙚 for SMART videos in Math, Science & Programming: http://bit.ly/SocraticaSubscribe ▶️ 𝙋𝙇𝘼𝙔𝙇𝙄𝙎𝙏𝙎: Study Tips http://bit.ly/StudyTipsPlaylist Python http://bit.ly/PythonSocratica Chemistry http://bit.ly/Chemistry_Playlist Calculus http://bit.ly/CalculusSocratica Geometry http://bit.ly/GeometrySocratica #2DNormalDistributions #math #MeanVector