--- title: "Joint Distributions, Covariance and Independence" description: "Two variables have a joint distribution, from which marginals follow. Covariance measures how they move together, and independence forces zero covariance without the converse holding." canonical: https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF source: https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Joint Distributions, Covariance and Independence Two variables have a joint distribution, from which marginals follow. Covariance measures how they move together, and independence forces zero covariance without the converse holding. Subject: Mathematics · Area: Probability · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF ## Ready when they can - Find marginal distributions from a joint table or density - Compute a covariance and interpret its sign - Give dependent variables with zero covariance ## Lesson: Two variables moving together A joint table holds both variables at once, and a marginal sums the other one out. Take P(X=0,Y=0)=0.4, P(X=0,Y=2)=0.1, P(X=1,Y=0)=0.2, P(X=1,Y=2)=0.3. Then P(X=1) adds the X=1 column: 0.2 + 0.3 = 0.5. And P(Y=2) adds the Y=2 row: 0.1 + 0.3 = 0.4. Test independence by factorising: P(X=0) times P(Y=0) is 0.5 times 0.6 = 0.3, but the joint cell holds 0.4, so these two are dependent. **Example.** Covariance averages (X minus mean) times (Y minus mean), with the shortcut Cov = E[XY] minus E[X]E[Y]. Here E[X] = 0.5 and E[Y] = 0.8. Only cell (1, 2) feeds E[XY]: 1 times 2 times 0.3 = 0.6. The means multiply to 0.5 times 0.8 = 0.4. So Cov = 0.6 minus 0.4 = 0.2. The positive sign says above average X tends to arrive with above average Y. That sign is direction only: it proves no cause and measures no strength. Independence forces zero covariance, but zero never proves independence. Take X symmetric about zero with Y = X squared: deviations cancel by symmetry, so the covariance is zero, yet Y is fully set by X. Lee's claim that zero covariance certifies independence breaks on this one example. Independence is the strong demand that every joint chance factorises into its marginals. **Tip.** Run the quick dependence check: compare one joint cell against the product of its marginals. Any mismatch certifies dependence. And never read a zero covariance as a clean bill of independence. **Recap.** Marginals sum the joint table out, covariance signs the co-movement, and zero covariance never certifies independence. ## Practice 13 questions on this page, each with its working shown. ## Needs first - [Expectation and Variance by Integration](https://lightmysky.com/learn/mathematics/expectation-and-variance-by-integration-mt_kRSaKgp3bs) - [Correlation and the Product-Moment Coefficient](https://lightmysky.com/learn/mathematics/correlation-and-the-product-moment-coefficient-mt_Rs_Z0P7ELJ) - [Double Integrals over General Regions](https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw) ## Opens up - [The Law of Large Numbers and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp) - [Brownian Motion and Its Defining Properties](https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z) - [Concentration Bounds and a High-Probability Guarantee](https://lightmysky.com/learn/computing/concentration-bounds-and-a-high-probability-guarantee-mt_WIltjK12oW)