Multivariate Distribution

Study guide: Exam P-1

A multivariate distribution describes the joint probabilistic behavior of two or more Random Variables. For random variables $X_1, X_2, \ldots, X_n$ , the joint distribution is characterized by the Joint Probability Function (discrete) or joint density function (continuous).

Marginal Probability Functions: distributions of individual variables after integrating/summing out the others
Conditional Probability Functions: distributions of one variable given fixed values of the others
Covariance and Correlation Coefficient: measures of linear dependence between pairs of variables
If all variables are Independent Random Variables, the joint distribution factors as the product of the marginals

Joint vs. Marginal Distribution $X$ and $Y$ each take values $\{0, 1\}$ with joint PMF: $P(0,0)=0.1$ , $P(0,1)=0.4$ , $P(1,0)=0.3$ , $P(1,1)=0.2$ .

Answer Marginal of $X$ : $P(X=0) = 0.1+0.4 = 0.5$ , $P(X=1) = 0.3+0.2 = 0.5$ . Marginal of $Y$ : $P(Y=0) = 0.1+0.3 = 0.4$ , $P(Y=1) = 0.4+0.2 = 0.6$ . Since $P(0,0) = 0.1 \neq P(X=0)P(Y=0) = 0.20$ , $X$ and $Y$ are not independent.