Hi guys, Welcome to the Advanced Video Search Engine, and you are watching A Finite Game of Infinite Rounds #SoME2. That video is uploaded by Joseph Newton at 2022-08-15T16:35:23-07:00. We are promoting this video only for entertainment and educational purpose only. So, I hope you like & enjoy it. [LUCvSsx6-EU]
| Name | A Finite Game of Infinite Rounds #SoME2 |
| Video Uploader | Video From Joseph Newton |
| Upload Date | This Video Uploaded At 15-08-2022 23:35:23 |
| Video Discription | A short video about a random variable with no expected value. Made for the Summer of Maths Exposition 2. 0:00 Let's play a game 2:33 A better-behaved example 4:49 Working through the maths 7:08 Does the game always finish? 9:34 Discussion, and another example 11:04 A challenge problem Correction: the sum at 4:32 should be P(1)+2P(2)+3P(3)+… Some extra details I found later after some discussion in the comments: The Cauchy distribution has a cool property that intuitively explains why it behaves so weirdly: the probability distribution of the average of a number independent Cauchy-distributed variables is the exact same Cauchy distribution! No matter how many runs you average, the distribution doesn't narrow, making the Cauchy distribution a counterexample to the central limit theorem. Why do the simulations at the start of the video seem to go up logarithmically? Let's say you run N games and average the results. The mean is infinite, but we could shoddily get around this by assuming no game runs for K rounds or longer, the probability of which is (1-1/K)^N. Intuitively K should vary with N, so let's fix this to some chosen threshold probability p (e.g. p=0.05) and rearrange to get K=1/(1-p^(1/N)). We can now calculate the expected value only summing up to the K-th term, and after approximating the harmonic numbers with a logarithm and using a small Taylor expansion I got an expected value of ln(N) - ln(-ln(p)) + γ - 1. So yes, by some weird metric it is logarithmic with the number of games! However I should mention that I actually rerecorded the programs a couple of times before I got results that fit with the flow of the video; in reality the average tends to jump around a lot more. LUCvSsx6-EU |
| Category | Education |
| Tags | Education Download MP4 | Education Download MP3 | Education Download MP4 360p | Education Download MP4 480p | Education Download MP4 720p | Education Download MP4 1080p |
