2D Random Variables

Checkpoints

• Covariance and Correlation Essential The covariance Cov(X, Y) = E[(X − E[X])(Y − E[Y])] measures the joint linear variability of two random variables, and its standardised form ρ(X, Y) = Cov(X, Y) / (σ_X σ_Y) is the Pearson correlation coefficient lying in [−1, 1]. This checkpoint derives the bilinearity of covariance, the variance-of-a-sum identity Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y), the Cauchy–Schwarz bound for correlation, and warns that zero covariance does not imply independence.
• Independence of Random Variables Essential Two random variables X and Y are independent when their joint distribution factorises as the product of the marginals — F(x, y) = F_X(x) F_Y(y), or equivalently the joint density (or PMF) factorises. This checkpoint gives the definition, proves the factorisation E[XY] = E[X] E[Y] for independent integrable variables, and explains the difference between pairwise and mutual independence for collections of more than two variables.
• Jointly Distributed Random Variables Essential A pair (X, Y) of random variables on the same probability space induces a joint distribution on ℝ²; the marginals are recovered by integrating out the other coordinate. This checkpoint defines joint CDF, joint PMF, and joint density, derives the marginal formulas from the joint law, and explains why the joint distribution carries strictly more information than the marginals alone.