# Isye 2027

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Unformatted text preview: d, then F (b, d) − F (b, c) − F (a, d) + F (a, c) ≥ 0 • limu→−∞ F (u, v ) = 0 for each v , and limv→−∞ F (u, v ) = 0 for each u • limu→∞ limv→∞ F (u, v ) = 1 It can be shown that any function F satisfying the above properties is the joint CDF for some pair of random variables (X, Y ). 4.2 Joint probability mass functions If X and Y are each discrete-type random variables on the same probability space, they have a joint probability mass function (joint pmf), denoted by pX,Y . It is deﬁned by pX,Y (u, v ) = P {X = u, Y = v }. If the numbers of possible values are small, the joint pmf can be easily described in a table. The joint pmf determines the probabilities of any events that can be expressed as conditions on X and Y. In particular, the pmfs of X and Y can be recovered from the joint pmf, using the law of total probability: pX (u) = pX,Y (u, v ) v pY (v ) = pX,Y (u, v ). u In this case, pX and pY are called the marginal pmfs of the joint pmf, pX,Y . The conditional pmfs are also determined by the joint pmf. For example, the conditi...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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