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A First Course in Probability and Markov Chains by Laura Poggiolini, Giuseppe Modica

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4

Vector valued random variables

4.1   Joint distribution

Let X and Y be two random variables. The distribution vector Images does not bear enough information about the relations occurring between X and Y, for instance to compute Images 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 Images. Let Images, Images and let (p1, . . ., pn) and (q1, . . ., qm) be the mass density vectors of Images and Images, respectively.

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

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