Week8 - Joint Distribution of two or More Random Variables Sometimes more than one measurement(r.v is taken on each member of the sample space In

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week 8 1 Joint Distribution of two or More Random Variables Sometimes more than one measurement (r.v.) is taken on each member of the sample space. In cases like this there will be a few random variables defined on the same probability space and we would like to explore their joint distribution. Joint behavior of 2 random variable (continuous or discrete), X and Y determined by their joint cumulative distribution function n – dimensional case ( ) ( ) . , , , y Y x X P y x F Y X = ( ) ( ) . , , ,..., 1 1 1 ,..., 1 n n n X X x X x X P x x F n = K

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week 8 2 Discrete case Suppose X , Y are discrete random variables defined on the same probability space. •T h e joint probability mass function of 2 discrete random variables X and Y is the function p X,Y ( x,y ) defined for all pairs of real numbers x and y by For a joint pmf p X,Y ( x,y ) we must have: p X,Y ( x,y ) 0 for all values of x,y and ( ) ( ) y Y and x X P y x p Y X = = = , , ( ) 1 , , = ∑∑ xy Y X y x p
week 8 3 Example for illustration Toss a coin 3 times. Define, X : number of heads on 1st toss, Y : total number of heads. The sample space is ={TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}. We display the joint distribution of X and Y in the following table Can we recover the probability mass function for X and Y from the joint table? To find the probability mass function of X we sum the appropriate rows of the table of the joint probability function. Similarly, to find the mass function for Y we sum the appropriate columns.

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week 8 4 Marginal Probability Function •T h e marginal probability mass function for X is h e marginal probability mass function for Y is Case of several discrete random variables is analogous. If X 1 ,…,X m are discrete random variables on the same sample space with joint probability function The marginal probability function for X 1 is The 2-dimentional marginal probability function for X 1 and X 2 is ( ) ( ) = y Y X X y x p x p , , () ( ) = x Y X Y y x p y p , , ( ) ( ) m m m X X x X x X P x x p n = = = ,..., ,... 1 1 1 ,... 1 ( ) = m n x x m X X X x x p x p ,..., 1 ,... 1 2 1 1 ,... ( ) ( ) = m n x x m X X X X x x x x p x x p ,..., 3 2 1 ,... 2 1 3 1 2 1 ,..., , , ,
week 8 5 Example Roll a die twice. Let X : number of 1’s and Y : total of the 2 die. There is no available form of the joint mass function for X , Y. We display the joint distribution of X and Y with the following table.

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This note was uploaded on 09/22/2010 for the course STATISTICS STA257 taught by Professor Mcdunnough during the Fall '10 term at University of Toronto- Toronto.

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Week8 - Joint Distribution of two or More Random Variables Sometimes more than one measurement(r.v is taken on each member of the sample space In

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