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Review_notes - 1 Review - conditional probability and...

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Unformatted text preview: 1 Review - conditional probability and conditional expectation 1.1 Conditional probability & distributions We consider two random variables (rv&s) X and Y with joint probability mass function (pmf) or joint density function (pdf) f X;Y ( x;y ) . It is known that & the marginal pmf or pdf of X f X ( x ) = Z all y f X;Y ( x;y ) dy , or f X ( x ) = X all y f X;Y ( x;y ) . & the marginal pmf or pdf of Y f Y ( y ) = Z all x f X;Y ( x;y ) dx , or f Y ( y ) = X all x f X;Y ( x;y ) . & the conditional pmf or pdf of X j Y = y is f X j Y ( x j y ) = f X;Y ( x;y ) f Y ( y ) . Remark 1 If the rv&s X and Y are independent, f X;Y ( x;y ) = f X ( x ) f Y ( y ) . As an immediate consequence, we have f X j Y ( x j y ) = f X ( x ) f Y ( y ) f Y ( y ) = f X ( x ) . Using the conditional distribution of X j Y , the marginal pmf or pdf of X can be expressed as f X ( x ) = Z all y f X j Y ( x j y ) f Y ( y ) dy , or f X ( x ) = X all y f X j Y ( x j y ) f Y ( y ) . (1) Often f X j Y ( x j y ) is a known parametric probability distribution with a given set of parameters. Under the represen- tation (1), the marginal distribution of X is called a mixed distribution (or mixture) . Example 2 ( two-point mixture ) Assume that Y is a discrete r.v. with 2 possible values, namely y 1 and y 2 , with f Y ( y 1 ) = p = 1 ¡ f Y ( y 2 ) . Then, f X ( x ) = f X j Y ( x j y 1 ) f Y ( y 1 ) + f X j Y ( x j y 2 ) f Y ( y 2 ) = p f X j Y ( x j y 1 ) + (1 ¡ p ) f X j Y ( x j y 2 ) . 1 Example 3 ( negative binomial ) Suppose that & X j & = & is Poisson distributed with mean & & & is gamma distributed with pdf f & ( & ) = ¡ ( ¡& ) & & 1 e & ¡¢ ¡ ( ¢ ) , & > , where ¢;¡ > with ¡ ( ¢ ) = Z 1 & & & 1 e & ¢ d& . Aside: Using integration by parts, ¡ ( ¢ + 1) = ¢ ¡ ( ¢ ) , for ¢ > . It follows that ¡ ( ¢ + n ) = ( ¢ + n ¡ 1) ¡ ( ¢ + n ¡ 1) = ( ¢ + n ¡ 1) ( ¢ + n ¡ 2) ¡ ( ¢ + n ¡ 2) ::: For ¢ a positive integer, ¡ ( ¢ + 1) = ¢ ¡ ( ¢ ) = ¢ ( ¢ ¡ 1) ¡ ( ¢ ¡ 1) = ¢ ( ¢ ¡ 1) ::: (1) ¡ (1) = ¢ ! Find the marginal distribution of X . f X ( x ) = Z 1 e & ¢ & x x ! ¡ ( ¡& ) & & 1 e & ¡¢ ¡ ( ¢ ) d& = ¡ & ¡ ( ¢ ) 1 x ! Z 1 & x + & & 1 e & ( ¡ +1) ¢ d& = ¡ & ¡ ( ¢ ) 1 x ! ¡ ( x + ¢ ) ( ¡ + 1) x + & Z 1 ( ¡ + 1) x + & & x + & & 1 e & ( ¡ +1) ¢ ¡ ( x + ¢ ) d& | {z } =1 = ¡ ( x + ¢ ) ¡ ( ¢ ) 1 x ! ¡ & ( ¡ + 1) x + & = ¡ ( x + ¢ ) ¡ ( ¢ ) 1 x ! & ¡ ¡ + 1 ¡ & & 1 ¡ + 1 ¡ x = & x + ¢ ¡ 1 x ¡& ¡ ¡ + 1 ¡ & & 1 ¡ + 1 ¡ x . With this pmf, X is known to be a negative binomial rv. 1.2 Conditional expectation In this section, we assume that X j Y = y is a continuous rv with a valid pdf f X j Y ( ¢j y ) (if X j Y is discrete, replace all the integral signs by summation signs). The conditional moment of X j Y is given by E [ X j Y = y ] = Z all x xf X j Y ( x j y ) dx ....
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This note was uploaded on 02/02/2010 for the course ACTSC 432 taught by Professor Davidlandriault during the Spring '09 term at Waterloo.

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Review_notes - 1 Review - conditional probability and...

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