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Probability Distributions

受眾: University類別: Math

簡介

Introduces three fundamental probability distributions — Normal, Binomial, and Poisson — with visual representations and key formulas. For the Normal distribution, the 68-95-99.7 empirical rule is animated by progressively shading regions under the bell curve.

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Probability Distributions

Description

Introduces three fundamental probability distributions — Normal, Binomial, and Poisson — with visual representations and key formulas. For the Normal distribution, the 68-95-99.7 empirical rule is animated by progressively shading regions under the bell curve.

Phases

# Phase Name Duration Description
1 Introduction 4s Title, brief description of what a probability distribution is
2 Normal Distribution 14s Bell curve, label μ and σ, animate 68%, 95%, 99.7% shading
3 Binomial Distribution 10s Bar chart for B(10, 0.4), formula, label n and p
4 Poisson Distribution 10s Bar chart for Pois(λ=3), formula, compare to Binomial shape
5 Comparison Summary 7s Side-by-side mini-displays of all three, key properties list
6 Applications 5s Brief text on where each distribution arises

Layout

+--------------------------------------------------+
|  Title: "Probability Distributions"             |
+--------------------------------------------------+
|                                                  |
|   NORMAL DISTRIBUTION (center)                   |
|   Bell curve with μ, σ labels                   |
|   Progressive shading:                           |
|   [68% fill] → [95% fill] → [99.7% fill]        |
|                                                  |
|   Then transitions to:                           |
|   BINOMIAL bars (B(10,0.4))                     |
|   Then: POISSON bars (λ=3)                      |
|                                                  |
|   Formula panel: right side throughout           |
+--------------------------------------------------+

Area Descriptions

  • Main axes (left 65%): Distribution plot — bell curve or bar chart
  • Right 35%: Distribution formula in MathTex, parameter labels
  • Below axes: Shading percentage labels (68%, 95%, 99.7%)

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

Notes

  • Normal PDF: (1/σ√(2π)) exp(-(x-μ)²/(2σ²)), use μ=0, σ=1
  • Binomial PMF: C(n,k) p^k (1-p)^(n-k), n=10, p=0.4
  • Poisson PMF: e^(-λ) λ^k / k!, λ=3
  • Shading done with axes.get_area() between x=-1 and x=1, etc.