{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapt 5b

# Chapt 5b - 5.5 Determining Probabilities from a Joint CDF...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ] &amp;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 ) &amp;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 ) &amp;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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 10

Chapt 5b - 5.5 Determining Probabilities from a Joint CDF...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online