Chapter3.pdf - Chapter 3 Multivariate Random Variables Joan Llull Probability and Statistics QEM Erasmus Mundus Master Fall 2015 joan.llull[at

Chapter3.pdf - Chapter 3 Multivariate Random Variables Joan...

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Chapter 3: Multivariate Random Variables Joan Llull Probability and Statistics. QEM Erasmus Mundus Master. Fall 2015 joan.llull [at] movebarcelona [dot] eu
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Joint and Marginal Distributions Chapter 3. Fall 2015 2
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Joint and marginal cdfs Multivariate random variable : a vector that includes several (scalar) random variables: X = ( X 1 ,...,X K ) 0 . Joint cdf : F X 1 ...X K ( x 1 ,...,x K ) P ( X 1 x 1 ,X 2 x 2 K x K ) Marginal cdf : F i ( x ) P ( X i x ) = P ( X 1 ≤ ∞ i x,. ..,X K ≤ ∞ ) = F ( ,...,x,. .., ) . Chapter 3. Fall 2015 3
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Joint pmfs and pdfs Joint pmf (discrete var.): P ( X 1 = x 1 ,X 2 = x 2 ,...,X K = x K ) . Joint pdf (continuous var.): a joint pdf f X 1 ...X K ( z 1 ,...,z K ) satis es: F X 1 ...X K ( x 1 ,...,x K ) Z x 1 -∞ ... Z x K -∞ f X 1 ...X K ( z 1 K ) dz 1 ...dz K . Properties of joint pdfs : f X 1 ...X K ( x 1 K ) 0 for all x 1 K . F X 1 ...X K ( ,..., ) = R -∞ ... R -∞ f X 1 ...X K ( z 1 K ) dz 1 ...dz K = 1 . P ( a 1 X 1 b 1 ,...,a K X K b k ) = R b 1 a 1 ... R b K a K f ( z 1 K ) dz 1 ...dz K . P ( X 1 = a 1 K = a K ) = 0 . P ( X 1 = a,a 2 X 2 b 2 K X K b K ) = 0 . (examples with discrete and continuous variables) Chapter 3. Fall 2015 4
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Marginal pmfs and pdfs Marginal pmf (discrete var.): P ( X i = x ) X x 1 ... X x K P ( X 1 = x 1 ,...X i = x,. ..,X K = x K ) . Marginal pdf (continuous var.): f i ( x ) = Z -∞ ... Z -∞ f X 1 ...X K ( z 1 ,...,x,. ..,z K ) dz 1 ...dx i - 1 dx i +1 ...dz K , or equivalently: F i ( x ) = Z x -∞ f i ( z ) dz. (examples with discrete and continuous variables) Chapter 3. Fall 2015 5
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Conditional Distributions and Independence Chapter 3. Fall 2015 6
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Conditional probability and independence Let A , B ⊂ F . The probability that A occurs given that B occurred, denoted by P ( A|B ) is formally de ned as: P ( ) P ( A∩B ) P ( B ) . Bayes' rule : P ( ) = P ( ) P ( B ) = P ( B|A ) P ( A ) P ( ) = P ( ) P ( B ) P ( A ) . A and B are independent if (the three below are equivalent): P ( ) = P ( A ) P ( ) = P ( B ) P ( ) = P ( A ) P ( B ) . Chapter 3. Fall 2015 7
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Conditional cdfs Let X be a random variable, and A an event, with P ( A ) 6 = 0 . The conditional cdf of X given A occurred is: F X |A ( x ) P ( X x ) = P ( X x ∩ A ) P ( A ) .
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