# lecture5 - ISYE 2028 A and B Lecture 5 Dr. Kobi Abayomi...

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ISYE 2028 A and B Lecture 5 Dr. Kobi Abayomi January 29, 2009 1 Joint Distributions Two given random variables X and Y have a general distribution — a joint distribution — that is an extension of the single variable deﬁnition and notation we generate from ﬁrst principles F X,Y ( x,y ) = P ( X x,Y y ) (1) Taking derivatives, dF X,Y ( x,y ), yields. . ...In the discrete case: P (( X,Y ) = ( x,y )) = p ( x,y ) (2) the joint probability mass function . ...In the continuous case P (( X,Y ) = ( x ± ±,y ± ± )) = f ( x,y ) (3) 1.1 Marginal Distributions We generate the marginal distributions for X and Y alone just as we did for contingency tables by summing over all values of the other variable. 1

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p x ( x ) = X y p ( x,y ); p y ( y ) = X x p ( x,y ) (4) f x ( x ) = Z R f ( x,v ) dv ; f y ( y ) = Z R f ( u,y ) dx (5) If you will recall our contingency table example(s), where we generated a marginal distribu- tion by summing over columns of the tale to yield the distributions of the margins. F X ( x ) = P ( X x ) = P ( X x,Y ≤ ∞ ) = P (lim y ↑∞ { X x,Y y } ) = lim y ↑∞ P ( { X x,Y y } ) = lim y ↑∞ F X,Y ( x,y ) The joint survival distribution can be generated from ﬁrst principles as well: P ( X > x,Y > y ) = = 1 - P ( { X > x,Y > y } c ) = 1 - P ( { X > x } c ∪ { Y > y } c ) ...by deMorgan’s laws. .. = 1 - P ( { X x } ∪ { Y y } ) ...by the inclusion-exclusion principle. .. 1 - [ P ( X x ) + P ( Y y ) - P ( X x,Y y )] ...by changing notation. .. = 1 - F X ( x ) - F Y ( y ) + F X,Y ( x,y ) You can use the facts above to verify: 2
P ( a X b,c Y d ) = F X,Y ( b,d ) + F X,Y ( a,c ) - F X,Y ( a,d ) - F X,Y ( b,c ) (6) This is just the fact that a positive area on the ( X,Y ) plane has a positive F X,Y volume above it if F is really a distribution function. This is in direct extension of the non-decreasing quality of F X is one dimension. 2 Examples 2.1 Example Take an experiment where we toss a coin three times. Let X the number of heads on the 1 st two tosses. Let Y the number of total heads. The sample space is Ω = { ( TTT ) ,..., ( HHH ) } each event with probability 1 / 8. The event space for the random variables is (

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## This note was uploaded on 11/08/2009 for the course ISYE 2028 taught by Professor Shim during the Spring '07 term at Georgia Institute of Technology.

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lecture5 - ISYE 2028 A and B Lecture 5 Dr. Kobi Abayomi...

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