06_01 - Jointly Distributed Random Variables Recall In...

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p. 6-1 Jointly Distributed Random Variables • Recall . In Chapters 4 and 5, focus on univariate random variable. However, often a single experiment will have more than one random variable which is of interest. R R R X 1 X 2 X n P X =( X 1 , … , X n ): R n . Definition. Given a sample space and a probability measure P defined on the subsets of , random variables X 1 , X 2 , … , X n : R are said to be jointly distributed . We can regard n jointly distributed r.v.’s as a random vector X 1 X 2 P A E A P X 1 ,X 2 P X 1 ,X 2 ( A ) =?? R 2 A occurs E A occurs P X 1 ,X 2 ( A )= P ( E A ) ( X 1 , X 2 ) Q : For A R n , how to define the probability of { X A } from P ? p. 6-2 For A R n , For A i R , i =1, …, n , P X 1 ,...,X n ( X 1 A 1 , ··· ,X n A n ) = P ( { ω | X 1 ( ω ) A 1 } { ω | X n ( ω ) A n } ) P X 1 ,...,X n ( A ) = P ( { ω | ( X 1 ( ω ) ,...,X n ( ω )) A } ) X 1 X 2 Definition. The probability measure of X ( P X , defined on R n ) is called the joint distribution of X 1 , …, X n . The probability measure of X i ( , defined on R ) is called the marginal distribution of X i . P X i Q : Why need joint distribution? Why are marginal distributions not enough? Example (Coin Tossing, LNp.4-2). X 2 : # of head on 1 st toss X 1 : total # of heads 0 (1/8) 1 (3/8) 2 (3/8) 3 (1/8) 0 (1/2) 1/8 [ 1/16 ] 2/8 [ 3/16 ] [ 3/16 ] 0 [ 1/16 ] 1 (1/2) 0 [ 1/16 ] [ 3/16 ] [ 3/16 ] [ 1/16 ]
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p. 6-3 blue numbers: joint distribution of X 1 and X 2 (black numbers): marginal distributions [read numbers]: joint distribution of another ( X 1 , X 2 ) Some findings: When joint distribution is given, its corresponding marginal distributions are known, e.g., P ( X 1 = i )= P ( X 1 = i , X 2 =0)+ P ( X 1 = i , X 2 =1), i =0, 1, 2, 3.
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shao-weicheng during the Fall '11 term at National Tsing Hua University, China.

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06_01 - Jointly Distributed Random Variables Recall In...

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