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Unformatted text preview: 5.5 Determining Probabilities from a Joint CDF 177 Determine the following: 1. Joint PMF of X and Y 2. Marginal PMF of X 3. Marginal PMF of Y . Solution The joint PMF is obtained from the relationship F XY ( a , b ) = summationdisplay x a summationdisplay y b p XY ( x , y ) . Thus, F XY ( 1 , 1 ) = 1 / 8 = p XY ( 1 , 1 ) F XY ( 1 , 2 ) = p XY ( 1 , 1 ) + p XY ( 1 , 2 ) = 5 / 8 p XY ( 1 , 2 ) = 5 / 8 1 / 8 = 1 / 2 F XY ( 2 , 1 ) = p XY ( 1 , 1 ) + p XY ( 2 , 1 ) = 1 / 4 p XY ( 2 , 1 ) = 1 / 4 1 / 8 = 1 / 8 F XY ( 2 , 2 ) = p XY ( 1 , 1 ) + p XY ( 1 , 2 ) + p XY ( 2 , 1 ) + p XY ( 2 , 2 ) = 1 p XY ( 2 , 2 ) = 1 / 4 The joint PMF becomes p XY ( x , y ) = 1 8 x = 1, y = 1 1 2 x = 1, y = 2 1 8 x = 2, y = 1 1 4 x = 2, y = 2 The marginal PMF of X is given by p X ( x ) = braceleftbigg p XY ( 1 , 1 ) + p XY ( 1 , 2 ) = 5 / 8 x = 1 p XY ( 2 , 1 ) + p XY ( 2 , 2 ) = 3 / 8 x = 2 The marginal PMF of Y is given by p Y ( y ) = braceleftbigg p XY ( 1 , 1 ) + p XY ( 2 , 1 ) = 1 / 4 y = 1 p XY ( 1 , 2 ) + p XY ( 2 , 2 ) = 3 / 4 y = 2 trianglesolid 178 Chapter 5 Multiple Random Variables 5.6 Conditional Distributions Recall that for two events A and B , the conditional probability of event A given event B is defined by P [ A  B ]= P [ A B ] P [ B ] which is defined when P [ B ] &gt; 0. In this section we extend the same concept to two random variables X and Y governed by a joint CDF F XY ( x , y ) . 5.6.1 Conditional PMF for Discrete Random Variables Consider two discrete random variables X and Y with the joint PMF p XY ( x , y ) . The conditional PMF of Y , given X = x , is given by p Y  X ( y  x ) = P [ X = x , Y = y ] P [ X = x ] = p XY ( x , y ) p X ( x ) provided p X ( x ) &gt; 0. Similarly, the conditional PMF of X , given Y = y , is given by p X  Y ( x  y ) = P [ X = x , Y = y ] P [ Y = y ] = p XY ( x , y ) p Y ( y ) provided p Y ( y ) &gt; 0. If X and Y are independent random variables, p X  Y ( x  y ) = p X ( x ) and p Y  X ( y  x ) = p Y ( y ) . Also, the conditional CDFs are defined by F X  Y ( x  y ) = P [ X x  Y = y ] = summationdisplay u x p X  Y ( u  y ) F Y  X ( y  x ) = P [ Y y  X = X ] = summationdisplay v y p Y  X ( v  x ) Example 5.8 The joint PMF of two random variables X and Y is given by p XY ( x , y ) = braceleftBigg 1 18 ( 2 x + y ) x = 1 , 2; y = 1 , 2 otherwise 5.6 Conditional Distributions 179 a. What is the conditional PMF of Y given X ? b. What is the conditional PMF of X given Y ? Solution From Example 5.2 we know that the marginal PMFs are given by p X ( x ) = summationdisplay y p XY ( x , y ) = 1 18 2 summationdisplay y = 1 ( 2 x + y ) = 1 18 ( 4 x + 3 ) x = 1 , 2 p Y ( y ) = summationdisplay x p XY ( x , y ) = 1 18 2 summationdisplay x = 1 ( 2 x + y ) = 1 18 ( 2 y + 6 ) y = 1 , 2 Thus, the conditional PMFs are given by p X  Y ( x  y ) = p XY ( x , y ) p Y ( y ) = 2 x + y 2 y + 6 p...
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 Fall '07
 Carlton
 Variance, Probability theory, Ann, joint CDF

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