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Unformatted text preview: § 9 Conditional distribution § 9.1 Introduction 9.1.1 Definitions. Let A be any event. Let X,X 1 ,X 2 ,... be random variables. Then (a) the conditional distribution of X given A has the cdf F ( x  A ) = P ( X ≤ x  A ); (b) the conditional joint distribution of X 1 ,...,X n given A has the joint cdf F ( x 1 ,...,x n  A ) = P ( X 1 ≤ x 1 ,...,X n ≤ x n  A ); (c) the conditional mass function of a discrete X given A is f ( x  A ) = P ( X = x  A ); (d) the conditional joint mass function of discrete X 1 ,...,X n given A is f ( x 1 ,...,x n  A ) = P ( X 1 = x 1 ,...,X n = x n  A ); (e) the conditional pdf of a continuous X given A is f ( x  A ), such that F ( x  A ) = Z x∞ f ( u  A ) du ; (f) the conditional joint pdf of continuous X 1 ,...,X n given A is f ( x 1 ,...,x n  A ), such that F ( x 1 ,...,x n  A ) = Z x 1∞ ··· Z x n∞ f ( u 1 ,...,u n  A ) du n ··· du 1 . 9.1.2 By conditioning on an event A , we limit our scope about the random variable(s) to those outcomes that are possible under the occurrence of A . Example. Let X = no. of casualties at a road accident. Conditional distributions of X given that the accident occurs in different countries may be very different, i.e. distribution of X { China } may be different from distribution of X { Sweden } , say. In fact, even the “overall” distribution of X is itself a “conditional” distribution, i.e. the distribution of X { earth } , if we think of the “earth” as part of a bigger “universe”. 60 9.1.3 Distributional concepts about random variables learned previously all stem from probability functions P ( · ). Analogous concepts can be obtained for conditional distributions by rebuilding everything using P ( · A ) (which is ALSO a probability function) in place of P ( · ). In fact, the usual P ( · ) can be written in general as P ( · Ω). This means that we can think of everything as being “conditional” on some event. § 9.2 Survey of results 9.2.1 CONDITIONAL INDEPENDENCE X 1 ,X 2 ,... are conditionally independent given A iff P ( X 1 ≤ x 1 ,...,X n ≤ x n  A ) = n Y i =1 P ( X i ≤ x i  A ) for all x 1 ,...,x n ∈ (∞ , ∞ ) and any n ∈ { 1 , 2 ,... } . 9.2.2 CONDITIONAL EXPECTATION E [ X  A ] = (DISCRETE) X x ∈ X (Ω) x P ( X = x  A ) , (CONTINUOUS) Z xf ( x  A ) dx. All properties of E [ · ] still hold for E [ ·  A ]: (i) X ≥ 0 given A ⇒ E [ X  A ] ≥ 0. (ii) (DISCRETE) E [ g ( X 1 ,...,X n )  A ] = X x 1 ∈ X 1 (Ω) ··· X x n ∈ X n (Ω) g ( x 1 ,...,x n ) P ( X 1 = x 1 ,...,X n = x n  A ) . (CONTINUOUS) E [ g ( X 1 ,...,X n )  A ] = Z x 1 ··· Z x n g ( x 1 ,...,x n ) f ( x 1 ,...,x n  A ) dx n ··· dx 1 ....
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 Fall '08
 SMSLee
 Probability

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