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Joint_Distribution - Joint and Continuous In many...

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Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 1 Joint Distributions—Discrete and Continuous In many statistical investigations, one is frequently interested in studying the relationship between two or more random variables, such as the relationship between annual income and yearly savings per family or the relationship between occupation and hypertension.
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Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 2 Discrete Variables For two discrete random variables X 1 and X 2 , the probability that X 1 will take the value x 1 and X 2 will take the value x 2 is written as P ( X 1 = x 1 , X 2 = x 2 ). Consequently, P ( X 1 = x 1 , X 2 = x 2 ) is the probability of the intersection of the events X 1 = x 1 and X 2 = x 2 . If X 1 and X 2 are discrete random variables, the function given by f ( x 1 , x 2 ) = P ( X 1 = x 1 , X 2 = x 2 ) for each pair of values ( x 1 , x 2 ) within the range of X 1 and X 2 is called the joint probability distribution or joint density function of X 1 and X 2 .
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Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 3 The distribution of probability is specified by listing the probabilities associated with all possible pairs of values x 1 and x 2 , either by formula or in a table . A function of two variables can serve as the joint probability distribution of a pair of discrete random variables X 1 and X 2 if and only if its values, f ( x 1 , x 2 ), satisfy the conditions 1. f ( x 1 , x 2 ) 0 for each pair of values ( x 1 , x 2 ) within its domain; ∑ ∑ = 1 2 , 1 ) , ( . 2 2 1 x x x x f where the double summation extends over all possible pairs ( x 1 , x 2 ) within its domain.
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Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 4 Joint distribution function: If X 1 and X 2 are discrete random variables, the function given by ∑∑ = = 1 2 ) , ( ) , ( ) , ( 2 2 1 1 2 1 x s x t t s f x X x X P x x F for - < x 1 < , - < x 2 < where f ( s , t ) is the value of the joint probability distribution of X 1 and X 2 at ( s , t ), is called the joint distribution function, or the joint cumulative distribution of X 1 and X 2 .
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Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 5 Marginal distribution: If X 1 and X 2 are discrete random variables and f ( x 1 , x 2 ) is the value of their joint probability distribution at ( x 1 , x 2 ), the function given by = = = 2 ) , ( ) ( ) ( 2 1 1 1 1 1 x x x f x f x X P for each x 1 within the range of X 1 is called the marginal distribution of X 1 . Correspondingly, the function given by = = = 1 ) , ( ) ( ) ( 2 1 2 2 2 2 x x x f x f x X P for each x 2 within the range of X 2 is called the marginal distribution of X 2 .
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Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 6 Conditional probability distribution: Consistent with the definition of conditional probability of events when A is the event X 1 = x 1 and B is the event X 2 = x 2 , the conditional probability distribution of X 1 = x 1 given X 2 = x 2 is defined as 0 ) ( provided all for ) ( ) , ( ) | ( 2 2 1 2 2 2 1 2 1 1 = x f x x f x x f x x f Similarly, the conditional probability distribution of X 2 given X 1 = x 1 is defined as 0 ) ( provided all for ) ( ) , ( ) | ( 1 1 2 1 1 2 1 1 2 2 = x f x x f x x f
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