Uploaded September 2025 | Updated September 2026, 3 weeks ago
âïž ðŸð€ð£ð£ððð© ð¬ðð©ð ðªðš ð€ð£ ððŒððððð
patreon.com/socratica
We've discussed the one-dimensional Normal Distribution (the bell curve) in a previous video, so you're all experts now! Normal Distributions youtu.be/xlxaa9YhT6A
In this lesson, we extend the familiar Bell Curve into two dimensions. The 2D normal distribution â also called the Gaussian distribution in two variables â describes how data spreads when thereâs 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. Youâll 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, youâll see how the 2D case prepares us for the general multivariate normal distribution in any number of dimensions
âïž ðð€ðª ððð£ ððªð¢ð¥ ð©ð€ ðšððð©ðð€ð£ðš ð€ð ð©ðð ð«ðððð€ ððð§ð:
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
â¶ïž ððŒððŸð ðððð:
Normal Distributions youtu.be/xlxaa9YhT6A
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! ððŠ
ðœððð€ð¢ð ð€ðªð§ ððð©ð§ð€ð£ ð€ð£ ððð©ð§ðð€ð£:
patreon.com/socratica
ð ðð ð§ððð€ð¢ð¢ðð£ð (affiliate links):
The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow
amzn.to/4j9n0YP
The Art of Statistics: How to Learn from Data by David Spiegelhalter
amzn.to/3S9E46a
How to Be a Great Student (from Socratica!)
ebook: amzn.to/2Lh3XSP
paperback: amzn.to/3t5jeH3
ð¬ ðŸððð¿ððð:
Written & Produced by: Michael Harrison & Kimberly Hatch Harrison
Edited by: Alivia Brown and Megi Shuke
Music License from Soundstripe
Code: 6F6NQWRP2DBJBIQZ
ð ðŒðœððð ððð ðððððððŸðððð:
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 ð§ ðððŒðð ðððð with Socratica?
ðº ððªððšðð§ððð for SMART videos in Math, Science & Programming:
bit.ly/SocraticaSubscribe
â¶ïž ðððŒðððððð:
Study Tips bit.ly/StudyTipsPlaylist
Python bit.ly/PythonSocratica
Chemistry bit.ly/Chemistry_Playlist
Calculus bit.ly/CalculusSocratica
Geometry bit.ly/GeometrySocratica
#2DNormalDistributions #math #MeanVector
âïž ðŸð€ð£ð£ððð© ð¬ðð©ð ðªðš ð€ð£ ððŒððððð
patreon.com/socratica
We've discussed the one-dimensional Normal Distribution (the bell curve) in a previous video, so you're all experts now! Normal Distributions youtu.be/xlxaa9YhT6A
In this lesson, we extend the familiar Bell Curve into two dimensions. The 2D normal distribution â also called the Gaussian distribution in two variables â describes how data spreads when thereâs 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. Youâll 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, youâll see how the 2D case prepares us for the general multivariate normal distribution in any number of dimensions
âïž ðð€ðª ððð£ ððªð¢ð¥ ð©ð€ ðšððð©ðð€ð£ðš ð€ð ð©ðð ð«ðððð€ ððð§ð:
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
â¶ïž ððŒððŸð ðððð:
Normal Distributions youtu.be/xlxaa9YhT6A
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! ððŠ
ðœððð€ð¢ð ð€ðªð§ ððð©ð§ð€ð£ ð€ð£ ððð©ð§ðð€ð£:
patreon.com/socratica
ð ðð ð§ððð€ð¢ð¢ðð£ð (affiliate links):
The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow
amzn.to/4j9n0YP
The Art of Statistics: How to Learn from Data by David Spiegelhalter
amzn.to/3S9E46a
How to Be a Great Student (from Socratica!)
ebook: amzn.to/2Lh3XSP
paperback: amzn.to/3t5jeH3
ð¬ ðŸððð¿ððð:
Written & Produced by: Michael Harrison & Kimberly Hatch Harrison
Edited by: Alivia Brown and Megi Shuke
Music License from Soundstripe
Code: 6F6NQWRP2DBJBIQZ
ð ðŒðœððð ððð ðððððððŸðððð:
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 ð§ ðððŒðð ðððð with Socratica?
ðº ððªððšðð§ððð for SMART videos in Math, Science & Programming:
bit.ly/SocraticaSubscribe
â¶ïž ðððŒðððððð:
Study Tips bit.ly/StudyTipsPlaylist
Python bit.ly/PythonSocratica
Chemistry bit.ly/Chemistry_Playlist
Calculus bit.ly/CalculusSocratica
Geometry bit.ly/GeometrySocratica
#2DNormalDistributions #math #MeanVector
