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elemprob-fall2010-page24

elemprob-fall2010-page24 - X and the density of Y are...

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10 Joint distributions Given two discrete random variables X and Y , we can talk about the joint distribution or joint density: P ( X = x, Y = y ) . Here the comma means “and” and this is an abbreviation for P (( X = x ) ( Y = y )) . An example: Suppose we roll two dice, X is the number on the first die, and Y is the total. We say X and Y are independent if P ( X = x, Y = y ) = P ( X = x ) P ( Y = y ) for all x and y . Given the joint density, we can find the density of X and Y by summing: P ( X = x ) = y P ( X = x, Y = y ) , P ( Y = y ) = x P ( X = x, Y = y ) . If we are given the joint density of X and Y , then the density of
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Unformatted text preview: X and the density of Y are called the marginal densities or marginal distributions. If we know the joint density of X and Y , we can talk about the conditional distribution (density) of X given Y = y . This is the expression P ( X = x | Y = y ) = P ( X = x,Y = y ) P ( Y = y ) . Just as E g ( X ) = ∑ x g ( x ) P ( X = x ), we have E g ( X,Y ) = X x,y g ( x,y ) P ( X = x,Y = y ) . 24...
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  • Spring '10
  • ansan
  • Probability distribution, Probability theory, probability density function, joint density, Types of probability distributions

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