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# s5 - Stat 5101 Lecture Slides Deck 5 Charles J Geyer School...

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Stat 5101 Lecture Slides Deck 5 Charles J. Geyer School of Statistics University of Minnesota 1

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Joint and Marginal Distributions When we have two random variables X and Y under discussion, a useful shorthand calls the distribution of the random vector ( X, Y ) the joint distribution and the distributions of the random variables X and Y the marginal distributions . 2
Joint and Marginal Distributions (cont.) The name comes from imagining the distribution is given by a table Y grass grease grub red 1 / 30 1 / 15 2 / 15 7 / 30 X white 1 / 15 1 / 10 1 / 6 1 / 3 blue 1 / 10 2 / 15 1 / 5 13 / 30 1 / 5 3 / 10 1 / 2 1 In the center 3 × 3 table is the joint distribution of the variables X and Y . In the right margin is the marginal distribution of X . In the bottom margin is the marginal distribution of Y . 3

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Joint and Marginal Distributions (cont.) The rule for finding a marginal is simple. To obtain a marginal PMF/PDF from a joint PMF/PDF, sum or integrate out the variable(s) you don’t want. For discrete, this is obvious from the definition of the PMF of a random variable. f X ( x ) = Pr( X = x ) = X y f X,Y ( x, y ) f Y ( y ) = Pr( Y = y ) = X x f X,Y ( x, y ) To obtain the marginal of X sum out y . To obtain the marginal of Y sum out x . 4
Joint and Marginal Distributions (cont.) For continuous, this is a bit less obvious, but if we define f X ( x ) = Z f X,Y ( x, y ) dy We see that this works when we calculate expectations E { g ( X ) } = Z g ( x ) f X ( x ) dx = Z g ( x ) Z f X,Y ( x, y ) dy dx = ZZ g ( x ) f X,Y ( x, y ) dy dx The top line is the definition of E { g ( X ) } if we accept f X as the PDF of X . The bottom line is the definition of E { g ( X ) } if we accept f X,Y as the PDF of ( X, Y ). They must agree, and do. 5

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Joint and Marginal Distributions (cont.) Because of non-uniqueness of PDF — we can redefine on a set of probability zero without changing the distribution — we can’t say the marginal obtained by this rule is the unique marginal, but it is a valid marginal. To obtain the marginal of X integrate out y . To obtain the marginal of Y integrate out x . 6
Joint and Marginal Distributions (cont.) The word “marginal” is entirely dispensable, which is why we haven’t needed to use it up to now. The term “marginal PDF of X ” means exactly the same thing as the the term “PDF of X ”. It is the PDF of the random variable X , which may be redefined on sets of probability zero without changing the distribution of X . “Joint” and “marginal” are just verbal shorthand to distinguish the univariate distributions (marginals) from the bivariate distri- bution (joint). 7

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Joint and Marginal Distributions (cont.) When we have three random variables X , Y , and Z under dis- cussion, the situation becomes a bit more confusing. By summing or integrating out one variable we obtain any of three bivariate marginals f X,Y , f X,Z , or f Y,Z . By summing or integrating out two variables we obtain any of three univariate marginals f X , f Y , or f Z .
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s5 - Stat 5101 Lecture Slides Deck 5 Charles J Geyer School...

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