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L11posted_rev1 - MULTIVARIATE DISTRIBUTIONS DATA Rarely...

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MULTIVARIATE DISTRIBUTIONS, DATA Rarely only one random variable. Usually many { X 1 , . . . , X n } & func- tion u ( X 1 , . . . , X n ). Need joint distribution. n = 2 . Notation { X, Y } . Joint p.m.f. f ( x, y ) = P ( X = x and Y = y ) P ( X = x, Y = y ) 1
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Exercise 2.6-3, p 140 x 1 2 3 1 0.05 0.15 0.20 y 2 0.10 0.10 0.10 3 0.15 0.15 0.00 f ( x, y ) 0 x y f ( x, y )= 1 P (( X, Y ) A ) = ∑ ∑ ( x,y ) A f ( x, y ) P ( X + Y 3 ) = = 0 . 30 . 2
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Marginal probability mass function. f X ( x ) f 1 ( x ) = P ( X = x ) = X y f ( x, y ) f Y ( y ) f 2 ( y ) = P ( Y = y ) = X x f ( x, y ) x 1 2 3 1 0.05 0.15 0.20 0.40 y 2 0.10 0.10 0.10 0.30 3 0.15 0.15 0.00 0.30 0.30 0.40 0.30 Find P ( X 2) = either from joint or marginal. 3
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Conditional probability mass func- tion. f X | Y ( x | y ) = f ( x, y ) f Y ( y ) f Y | X ( y | x ) = f ( x, y ) f X ( x ) Notation differs from your text. Meaning. Fix value of one variable and take conditional probability. x 1 2 3 y = 1 Conditional probabilities act just like ordinary probabilities and means and variances can be computed with them. 4
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n random variables Joint p.m.f f ( x 1 , . . . , x n ) = P ( X 1 = x 1 , . . . , X n = x n ) f ( x 1 , . . . , x n ) 0 x 1 . . .
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