Lecture8 - Lecture 8 Multi-dimensional distributions 1. One...

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1 Lecture 8 Multi-dimensional distributions z z One may check that the following function are valid jcdf’s. That is they satisfy all the properties that a jcdf must posess. z Example: 1. 0 otherwise 2. 0 otherwise = ) , ( y x F XY 0 , 0 ), 1 )( 1 ( 2 y x e e y x = ) , ( y x F XY 5 3 , 1 0 , 2 3 y x y x Classification of Two-Dimensional Random Vectors z In analogy to random variables the random vectors may be classified into three categories: 1. Jointly continuous random vectors, These are random vectors the cdf’s of which contain no jumps. 2. Jointly discrete random vectors These are random vectors that take only discrete values, e.g. only integers Instead of using a jcdf to describe the distribution of such a vector we use the joint probability function (jpmf), which is defined as follows: Definition 5 Consider , where Then the jpmf of (X,Y) is defined as Clearly, if is a jcdf of (X,Y) and is its jpmf, then S ζ ,...}. ,... { ) ( ,...}, ,... , { ) ( 1 2 1 m n y y Y x x x X ,... 2 , 1 , ), , ( ) , ( = = = = j i y Y x X P j i P j i XY ) , ( y x F XY ) , ( j i P XY ∑∑ == = ] [ 1 ] [ 1 ), , ( ) , ( x i y j XY XY j i P y x F ) , ( Y X X =
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2 where denotes the integer part of , i.e. the biggest integer , such that 3. Mixed random vectors ] [ a ' a . ' a a a Joint Prob. Density function ± Much like the 1-D case, we have a density function corresponding to a 2-D distribution. Definition : The joint Prob. Density function (JPDF) of two random variables x and y, denoted by is defined as: ) , ( y x f XY y x y x F y x f XY XY = ) , ( ) , ( 2 2 ) , ( R y x z Remark: If one of the variables is discrete ,we then introduce a function. z Properties of JPDF : 1. The JPDF is nonnegative, 2. The JPDF integrates to 1, (.) δ 0 ) , ( y x f XY R y x , ∫∫ +∞ +∞ = 1 ) , ( dxdy y x f XY z Example:
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Lecture8 - Lecture 8 Multi-dimensional distributions 1. One...

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