lect16

# Lect16 - JOINT DISTRIBUTION FUNCTIONS INDEPENDENT RANDOM VARIABLES Outline JOINT DISTRIBUTION FUNCTIONS Joint Probability Mass Function Joint

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JOINT DISTRIBUTION FUNCTIONS Outline JOINT DISTRIBUTION FUNCTIONS Joint Probability Mass Function Joint Density Function INDEPENDENT RANDOM VARIABLES Independence: Discrete Case Independence: Continuous Case A Useful Criterion 1 / 18 Xinghua Zheng Lect 16: Joint distribution functions; Independence

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JOINT DISTRIBUTION FUNCTIONS Joint Probability Mass Function Joint Density Function JOINT DISTRIBUTION FUNCTIONS Often we’re interested in the relationship between random variables, in which case we need to model them jointly Suppose the random variables X , Y are deﬁned on a same sample space. Their joint cumulative distribution function (for short, joint cdf) is deﬁned by F ( x , y ) = P [ X x , Y y ] , -∞ < x , y < . How to get the (marginal) cdf’s of X and Y ? F X ( x ) = P [ X x ] = P [ X x , Y ] = F ( x , ) If Y can only take values b , then F X ( x ) = F ( x , ) = F ( x , b ) F Y ( y ) = P [ Y y ] = P [ X , Y y ] = F ( , y ) If X can only take values a , then F Y ( y ) = F ( , y ) = F ( a , y ) 2 / 18 Xinghua Zheng Lect 16: Joint distribution functions; Independence
JOINT DISTRIBUTION FUNCTIONS Joint Probability Mass Function Joint Density Function Joint Probability Mass Function When X and Y are both discrete random variables, it’s more convenient to work with the joint probability mass function (for short, joint pmf) deﬁned in the following way f ( x , y ) = P [ X = x , Y = y ] Based on the joint pmf we can compute the (marginal) pmf’s of X and Y as follows: f X ( x ) = P [ X = x ] = P [ X = x , Y ] = y f ( x , y ) f Y ( y ) = P [ Y = y ] = P [ X , Y = y ] = x f ( x , y ) 3 / 18 Xinghua Zheng Lect 16: Joint distribution functions; Independence

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JOINT DISTRIBUTION FUNCTIONS Joint Probability Mass Function Joint Density Function EXAMPLE: THINNING A POISSON PROCESS Suppose the number of customers arriving at a post ofﬁce follows a Poisson process with rate λ . Suppose further that at all times, the next incoming customer is male with probability p and is a female with probability q = 1 - p Let X = number of male customers before time 1; and Y = number of female customers before time 1 Their joint pmf? Marginal pmf’s? 4 / 18 Xinghua Zheng Lect 16: Joint distribution functions; Independence
JOINT DISTRIBUTION FUNCTIONS Joint Probability Mass Function Joint Density Function EXAMPLE: THINNING A POISSON PROCESS, ctd Joint pmf: P [ X = i , Y = j ] = P [ X + Y = i + j ] · P [ X = i , Y = j | X + Y = i + j ] = · = · .

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## This note was uploaded on 11/27/2011 for the course ISOM 3540 taught by Professor Zheu during the Spring '11 term at HKUST.

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Lect16 - JOINT DISTRIBUTION FUNCTIONS INDEPENDENT RANDOM VARIABLES Outline JOINT DISTRIBUTION FUNCTIONS Joint Probability Mass Function Joint

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