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Ch97_401 - Probability I(B 97 Chapter 6 Jointly Distributed...

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Probability I (B) 97 Chapter 6 : Jointly Distributed Random Variables National Taiwan Normal University. Lecturer: Pi-Wen 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 0 p ( x, y ) 1 , X all ( x,y ) p ( x, y ) = 1 P (( x, y ) A ) = X X ( x,y ) A p ( x, y ) 6-1
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Chapter 6: Jointly Distributed Random Variables 6-2 Example 6-1 (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 Ω = { (1 , 1) , (1 , 2) , · · · , (6 , 6) } p ( x, y ) = P ( X = x, Y = y ) Example 6-2 (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 6-3 . 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 } , -∞ < x, y < 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 ∂x∂y F ( x, y ) . where f ( x, y ) is called the (joint pdf) of X and Y . (graphic & 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
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Chapter 6: Jointly Distributed Random Variables 6-3 when D is the set { ( x, y ) : a x b, c y d } . One can show this holds when D is any set. Example 6-4 (HT 4.1-7) . p ( x, y ) = xy 2 30 , x = 1 , 2 , 3 y = 1 , 2 . Table the pmf function of X and Y . Find P (1 < X 3 , 1 Y < 2), P ( X 2 , Y 2) 1/6, 1/2 Example 6-5 (HT 4.1-9) . f X,Y ( x, y ) = 3 2 x 2 (1 - | y | ); - 1 < x < 1 , - 1 < y < 1 Let A = { ( x, y ) | 0 < x < 1 , 0 < y < x } , find P (( X, Y ) A ) (domain for the double integral) Example 6-6 (HT 4.1-10, 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 1 / 2 , 0 Y 1 / 2) = 1 / 4 Example 6-7 (Uniform on a square) . f X,Y ( x, y ) = 1 for 0 x 1 and 0 y 1 Find P ( | X - Y | ≤ 1 / 2) = 3 / 4
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