{"id":49264,"date":"2025-12-14T16:27:11","date_gmt":"2025-12-14T16:27:11","guid":{"rendered":"https:\/\/idea.membrane.com\/songwriter\/?p=49264"},"modified":"2025-12-14T16:27:11","modified_gmt":"2025-12-14T16:27:11","slug":"continuous-probability-distributions","status":"publish","type":"post","link":"https:\/\/kingarthur.com\/songwriter\/2025\/12\/14\/continuous-probability-distributions\/","title":{"rendered":"Continuous Probability Distributions"},"content":{"rendered":"

Continuous-Probability-Distributions.mp3<\/a>
\nContinuous-Probability-Distributions.mp4<\/a>
\nContinuous-Probability-Distributions-Pt-2.mp3<\/a>
\nContinuous-Probability-Distributions-Pt-2.mp4<\/a>
\nContinuous-Probability-Distributions-intro.mp3<\/a> <\/p>\n

[Intro]
\nThe integral
\n(Is integral)<\/p>\n

[Verse 1]
\nOh no, much to our regret
\n(This is not a finite set)
\nFor our summation
\n(Reveals deviation)<\/p>\n

[Bridge]
\nThe integral
\n(Is integral)<\/p>\n

[Chorus]
\nContinuous probability
\n(Distribution)
\nTime for us to see reality
\n(Calculation)<\/p>\n

[Verse 2]
\nOptimization
\n(And derivation)
\nIn the equation
\n(Showin’ our deviation)<\/p>\n

[Bridge]
\nThe integral
\n(Is integral)<\/p>\n

[Chorus]
\nContinuous probability
\n(Distribution)
\nTime for us to see reality
\n(Calculation)<\/p>\n

[Outro]
\nThe evolution
\n(Of our institution)
\nLeaving no solution
\nThe integral
\n(Is integral)
\nFor the people<\/p>\n

ABOUT THE SONG AND THE SCIENCE
\nThe Role of Calculus in Standard Deviation\u00a0Calculus comes into play in the mathematical theory underlying statistics:<\/p>\n

* Continuous Probability Distributions: When data is not a finite set of points but a continuous probability distribution (like the normal distribution, or “bell curve”), the summation (Sigma) in the standard deviation formula is replaced by an integral (int). This integral calculates the variance (and thus the standard deviation) for an infinite range of possible values.<\/p>\n

* Optimization and Derivation: The use of squared differences (variance) is preferred over absolute differences in statistics largely for calculus-based reasons. The sum of squared deviations is a smooth, continuous, and differentiable function, whereas the sum of absolute deviations is not .Differentiation is used to prove that the mean is the value that minimizes the sum of the squared deviations from all data points. This is a crucial property for developing efficient and optimal statistical estimators.<\/p>\n

* Locating Inflection Points: In a normal distribution graph, the standard deviation (sigma) corresponds precisely to the distance from the mean (mu) to the curve’s inflection points (where the curvature changes from concave-down to concave-up). Finding these inflection points is achieved by taking the second derivative of the probability density function and setting it to zero.<\/p>\n

In summary, while you use algebra for basic calculations, the reasons we define and use standard deviation the way we do are rooted in calculus, especially in advanced statistical theory and continuous distributions.<\/p>\n

From the album “Nonlinear<\/a>“<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Continuous-Probability-Distributions.mp3 Continuous-Probability-Distributions.mp4 Continuous-Probability-Distributions-Pt-2.mp3 Continuous-Probability-Distributions-Pt-2.mp4 Continuous-Probability-Distributions-intro.mp3 [Intro] The integral (Is integral) [Verse 1] Oh no, much to our regret (This is not a finite set) For our summation (Reveals deviation) [Bridge] The integral (Is integral) [Chorus] Continuous probability (Distribution) Time for us to see reality (Calculation) [Verse 2] Optimization (And derivation) In the equation (Showin’ our […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","footnotes":""},"categories":[3,5,9,14],"tags":[4190],"class_list":["post-49264","post","type-post","status-publish","format-standard","hentry","category-4d-music","category-chaos-theory","category-daniel","category-lyrics","tag-nonlinear"],"_links":{"self":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/49264","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/comments?post=49264"}],"version-history":[{"count":4,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/49264\/revisions"}],"predecessor-version":[{"id":49303,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/49264\/revisions\/49303"}],"wp:attachment":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/media?parent=49264"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/categories?post=49264"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/tags?post=49264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}