l21 - Chapter 5 Joint Distributions Single Random Variables...

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Chapter # 5 Joint Distributions Single Random Variables: Discrete Continuous Univariate

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Two Dimensional Random variables Discrete Continuous Bivariate
Discrete Joint Density: Let X and Y be discrete r.v, the ordered pair (X,Y) is called a two dimensional discrete r.v, a function ] y Y and x X [ P f XY = = = is called the joint density for (X,Y) and is a probability density/probability mass function for two random variables.

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Necessary and Sufficient Conditions : ∑∑ = 2200 x all Y all XY XY 1 ) y , x ( f . 2 0 ) y , x ( f . 1 Discrete Case:
Continuous Case: ∫∫ ∫ ∫ = = - - b a d c XY XY XY dx dy ) y , x ( f ] d Y c [ P and ] b X a [ P . 3 1 dx dy ) y , x ( f . 2 0 ) y , x ( f . 1 for a,b,c,d real is called the joint density for (X,Y)

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Note : In one dimensional continuous case, the probabilities correspond to areas whereas in the case of 2-D, it corresponds to volumes. Example: In an automobile plant two tasks are performed by robots. The first entails welding two joints; the second, tightening 3 bolts. Let X denote the number of defective welds and Y the number of improperly tightened bolts produced per car. Since X and Y are each discrete, (X,Y)
is a 2- dimensional discrete random Variable. Past data indicates that the joint

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l21 - Chapter 5 Joint Distributions Single Random Variables...

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