Covariance Illustrated: $\operatorname{Cov}(X,Y)=1$

Overview

A short animation that visualizes two random variables whose covariance equals 1. The viewer sees a scatter plot, the mean points, and a geometric interpretation of covariance as the average product of deviations, concluding with the numeric result 1.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Fade‑in title "Covariance = 1" and a blank coordinate plane.
2	Scatter Plot	~6s	Animate a set of points sampled from a bivariate normal distribution with covariance 1 (e.g., correlation ≈ 0.7). Show the points appearing one by one.
3	Means & Deviations	~5s	Highlight the mean point $(\bar x,\bar y)$. Draw thin arrows from each sample point to the mean, representing deviations $(x_i-\bar x,\,y_i-\bar y)$.
4	Covariance Calculation	~4s	Overlay a semi‑transparent rectangle for each point whose area equals the product $(x_i-\bar x)(y_i-\bar y)$. Use a running average label that updates to show the cumulative mean of these products, converging to 1.
5	Result Summary	~2s	Fade out the rectangles, keep the mean point and arrows, and display the final equation $\operatorname{Cov}(X,Y)=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)=1$.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                 MAIN (plot)                 │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional) – step title or result  │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Coordinate axes, scatter points, mean point, deviation arrows, rectangle overlays	Central visual focus; occupies most of the frame
Caption	Short step titles (e.g., "Scatter Plot", "Result") and final covariance equation	Small footer; appears only in phases 1, 2, and 5

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Notes

• Assumed a bivariate normal distribution with mean 0 and covariance matrix $[[1,1],[1,1]]$ to produce covariance 1.
• Number of sample points set to 30 for visual clarity while still demonstrating convergence.
• No persistent textual explanations beyond the footer; all concepts are shown visually.
• Total animation length ≈ 20 seconds, well under the 30‑second guideline.
• Single Scene class will contain all phases sequentially.