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Unformatted text preview: Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 2022, 2009 Example continue Consider a circle of radius R and chosen a point at random and denote X and Y to the coordinates of the point chosen. Identify the density function: f ( x,y ) = c, if x 2 + y 2 R 2 , if x 2 + y 2 > R 2 (a) Determine c . (b) Find the marginals. (c) Let D be the distance from the origin of the point selected, find P ( D d ) =? Example X Exponential( ) , Y = X 3 . Find F X,Y ( x,y ) =?. Conditioning and independence Definition Suppose P ( X = x ) > 0. The conditional distribution of Y , given X = x , is the probability distribution assigning probability P ( Y B, X = x ) P ( X = x ) . If X and Y are discrete P ( Y = y  X = x ) = P ( Y = y,X = x ) P ( X = x ) = P X,Y ( x,y ) P X ( x ) = P Y  X ( y  x ) . Continuous cases Definition The conditional density of Y given X = x is f Y  X ( y  x ) = f X,Y ( x,y ) f X ( x ) , f X ( x ) > , all y R 1 . Definition 2 P ( a Y b  X = x ) = Z b a f Y  X ( y  x ) dy. Theorem 3 P ( a X b, c Y d ) = Z d c Z b a f X ( x ) f Y  X ( y  x ) dxdy. Example: Discrete case P X,Y (0 , 0) = . 4 , P X,Y (0 , 1) = . 2 , P X,Y (1 , 0) = . 1 , P X,Y (1 , 1) = . 3 . Find P X  Y ( x  1) =?. Example: continuous case The joint density of X and Y is given by f ( x,y ) = 15 2 x (2 x y ) , < x < 1 , < y < 1; , otherwise. Find f X  Y ( x  y ). Bernoulli(1/2) example revisit X Bernoulli(1/2). Y 1 = X 1 , Y 2 = 1 X. P ( Y 1 x, Y 2 y ) =? Solution F Y 1 ,Y 2 ( x,y ) = , { y < } or { x < 1 } or { x + y < } or { x < 0 and y < 1 } ; 1 / 2 , { x + y 0 and x < 0 and y 1 } or { x + y 0 and x 0 and y < 1 } 1 , { x + y 0 and x 0 and y 1 } Conditioning and independence Definition Suppose P ( X = x ) > 0. The conditional distribution of Y , given X = x , is the probability distribution assigning probability P ( Y B, X = x ) P ( X = x ) . 1 If X and Y are discrete P ( Y = y  X = x ) = P ( Y = y,X = x ) P ( X = x ) = P X,Y ( x,y ) P X ( x ) = P Y  X ( y  x ) ....
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This note was uploaded on 11/15/2009 for the course STAT 311 taught by Professor Staff during the Fall '08 term at Wisconsin.
 Fall '08
 Staff
 Statistics

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