A First Course in Probability and Markov Chains

4

Vector valued random variables

4.1   Joint distribution

Let X and Y be two random variables. The distribution vector does not bear enough information about the relations occurring between X and Y, for instance to compute where A and B are generated by X and Y, respectively; one needs to introduce a new object called the joint (probability) distribution of X and Y.

Example 4.1 (Discrete random variables) Let X and Y be two discrete random variables on the same probability space . Let , and let (p₁, . . ., p_n) and (q₁, . . ., q_m) be the mass density vectors of and , respectively.

The image of the map (X, Y) : Ω → is a subset of {(x_i, y_j), i = 1, . . ., n, j = 1, . . ., m} (whose cardinality ...