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MathidiaMathematics Academy
Probability
Lesson 40 of 75
Review

Joint distributions

Objective. Analyse dependence, covariance and conditional distributions.

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  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

A joint distribution describes the probability behaviour of two random variables X and Y together, given by p(x, y) = P(X = x, Y = y) for discrete pairs, from which marginal distributions p(x) = Σᵧ p(x,y) recover each variable's individual behaviour. Covariance Cov(X,Y) = E(XY) − E(X)E(Y) measures whether X and Y tend to increase together (positive), move oppositely (negative), or show no linear association (zero), and dividing by the product of standard deviations gives the correlation coefficient. Conditional distributions p(y|x) = p(x,y)/p(x) describe Y's behaviour once X is fixed, and X and Y are independent precisely when p(x,y) = p(x)p(y) for all pairs. For two fair coin flips, if X is the first flip and Y the second, Cov(X,Y) = 0 since the flips are independent.

Worked example

Two random variables X and Y have E(X) = 2, E(Y) = 3 and E(XY) = 8. Find the covariance of this joint distribution.

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Illustrated card showing 2 … 3
  1. 1Recall the covariance formula Cov(X,Y) = E(XY) − E(X)E(Y).
  2. 2Compute the product E(X)E(Y) = 2 × 3.
  3. 3Simplify to get E(X)E(Y) = 6.
  4. 4Subtract from E(XY): Cov(X,Y) = 8 − 6.

Answer: Cov(X,Y) = 2

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Common mistake. Students assume Cov(X,Y) = 0 proves X and Y are independent; zero covariance only rules out linear association, and variables can be dependent yet uncorrelated, such as X and X² for symmetric X.

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