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lect16

# lect16 - JOINT DISTRIBUTION FUNCTIONS INDEPENDENT RANDOM...

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JOINT DISTRIBUTION FUNCTIONS INDEPENDENT RANDOM VARIABLES 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 INDEPENDENT RANDOM VARIABLES 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 INDEPENDENT RANDOM VARIABLES 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 INDEPENDENT RANDOM VARIABLES 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 INDEPENDENT RANDOM VARIABLES 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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• Spring '11
• Zheu
• Probability theory, probability density function, Cumulative distribution function, Joint Distribution Functions, Xinghua Zheng

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