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# lec11 - Compound p.m.f 1st processor 4001 4002 450 x x x x...

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Compound p.m.f. Motivation: Real problems very seldom concern a single random variable. As soon as more than 1 variable is involved it is not suﬃcient to think of modeling them only individually - their joint behavior is important. 2 variables case: Consider the two variables: X, Y are two discrete variables. The joint probability mass function is defined as p X,Y ( x, y ) := P ( X = x Y = y ) Example: A box contains 5 unmarked PowerPC G4 processors of different speeds: 2 400 mHz 1 450 mHz 2 500 mHz Select two processors out of the box (without replacement) and let X = speed of the first selected processor Y = speed of the second selected processor 1 Example (Cont’d) Summary: For a sample space we can draw a table of all the possible combinations of processors. We will distinguish between processors of the same speed by using the subscripts 1 and 2 1st processor Ω 400 1 400 2 450 500 1 500 2 400 1 - x x x x 400 2 x - x x x 2nd processor 450 x x - x x 500 1 x x x - x 500 2 x x x x - In total we have 5 · 4 = 20 possible combinations. Since we draw at random, we assume that each of the above combinations is equally likely. This yields the following probability mass function: 2 Example (Cont’d) Probabilities: 2nd processor mHz 400 450 500 400 0.1 0.1 0.2 1st proc.

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