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Unformatted text preview: Probability I (B) 97 Chapter 6 : Jointly Distributed Random Variables National Taiwan Normal University. Lecturer: PiWen Tsai Joint probability density function Jointly Distributed Random Variables We want to discuss collections of two random variables ( X,Y ), which are known as random vectors. The joint distribution of a pair of random variables X and Y is the probability distri bution over the plane defined by P ( D ) = P (( X,Y ) D ) for subsets D of the plane. Joint pmf In the discrete case, we can define the joint pmf as p ( x,y ) = P ( X = x,Y = y ) . Note that the comma means and (combined or joint outcome) . The range of the joint outcome ( X,Y ) is the set of all ordered pairs ( x,y ) with x in the range of X and y in the range of Y , and satisfying p ( x,y ) 1 , X all ( x,y ) p ( x,y ) = 1 P (( x,y ) A ) = X X ( x,y ) A p ( x,y ) 61 Chapter 6: Jointly Distributed Random Variables 62 Example 61 (Roll two fair dice) . Let X be the smaller and Y be the larger outcome on the dice. Write down the joint pmf of X and Y . Sample space = { (1 , 1) , (1 , 2) , , (6 , 6) } p ( x,y ) = P ( X = x,Y = y ) Example 62 (Roll two fair dice) . Let X be the sum of the two dice, and Y be the minimum. Table the pmf function of X and Y . Example 63 . Two different balls drawn at random without replacement. An urn with 3 red, 4 white and 5 blue balls and we randomly select 3 balls from the urn. Let X and Y , respectively, be the number of red and white balls chosen are randomly selected. Determine the joint probability mass function of X and Y . Joint distribution function of X and Y For any two RVs X and Y , the joint (cumulative) distribution function of X and Y is defined by F X,Y ( x,y ) = P { X x,Y y } , &lt; x,y &lt; In the continuous case, this is Z x Z y f X,Y ( x,y ) dy dx, and so we have f ( x,y ) = 2 xy F ( x,y ) . where f ( x,y ) is called the (joint pdf) of X and Y . (graphic &amp; properties) Probabilities of events determined by X and Y P ( a X b,c Y d ) = Z b a Z d c f X,Y ( x,y ) dy dx, or P (( X,Y ) D ) = Z Z D f X,Y dy dx Chapter 6: Jointly Distributed Random Variables 63 when D is the set { ( x,y ) : a x b,c y d } . One can show this holds when D is any set. Example 64 (HT 4.17) . p ( x,y ) = xy 2 30 , x = 1 , 2 , 3 y = 1 , 2 . Table the pmf function of X and Y . Find P (1 &lt; X 3 , 1 Y &lt; 2), P ( X 2 ,Y 2) 1/6, 1/2 Example 65 (HT 4.19) . f X,Y ( x,y ) = 3 2 x 2 (1  y  ); 1 &lt; x &lt; 1 , 1 &lt; y &lt; 1 Let A = { ( x,y )  &lt; x &lt; 1 , &lt; y &lt; x } , find P (( X,Y ) A ) (domain for the double integral) Example 66 (HT 4.110, Uniform on a triangle) . f X,Y ( x,y ) = 2 for 0 x y 1 Show that it is a pdf. Find P (0 X...
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 Spring '10
 Tom

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