An introduction to probability density functions Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/thanks Curious about measure theory? This does require some background in real analysis, but if you want to dig in, here is a textbook by the always-great Terence Tao. https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf Also, for the real analysis buffs among you, there was one statement I made in this video that is a rather nice puzzle. Namely, if the probabilities for each value in a given range (of the real number line) are all non-zero, no matter how small, their sum will be infinite. This isn't immediately obvious, given that you can have convergent sums of countable infinitely many values, but if you're up for it see if you can prove that the sum of any uncountable infinite collection of positive values must blow up to infinity. Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com/ Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown