Probability Distributions
対象: Universityカテゴリ: Math
説明
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.