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Unformatted text preview: Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 13-15, 2009 Example • Suppose we roll two fair six-sided dice, one red and one blue. – Let A be the event that two dice show the same value. – Let B be the event that the sum of the two dice is equal to 12. – Let C be the event that the red die shows 4. – Let D be the event that the blue die shows 4. Answer the following questions ¡2-¿ (a) Are A and B independent? (b) Are A and C independent? (c) Are A and D independent? (d) Are C and D independent? (e) Are A , C , and D all independent? The joint discrete distribution function Let X 1 ,X 2 ,...,X n be n discrete random variables all defined on the same probability space. The joint discrete distribution function of X 1 ,X 2 ,...,X n , denoted by p X 1 ,X 2 ,...,X n ( x 1 ,x 2 ,...,x n ), is the following function: • p X 1 ,X 2 ,...,X n : R n → R • p X 1 ,X 2 ,...,X n ( x 1 ,x 2 ,...,x n ) = P [ X 1 = x 1 ,X 2 = x 2 ,...,X n = x n ] Let X 1 ,X 2 ,...,X n be random variables associated with an experiment. Suppose that the sample space (i.e., the set of possible outcomes) of X i is the set R i . Then the joint random variable ¯ X = ( X 1 ,X 2 ,...,X n ) is defined to be the random variable whose outcomes consist of ordered n-tuples of outcomes, with the i th coordinate lying in the set R i . The sample space Ω of ¯ X is the Cartesian product of the R i ’s: Ω = R 1 × R 2 × ··· × R n . The joint distribution function of ¯ X is the function which gives the probability of each of the outcomes of ¯ X . Mutually independent random variables The random variables X 1 , X 2 , ..., X n are mutually independent if P ( X 1 = r 1 ,X 2 = r 2 ,...,X n = r n ) = P ( X 1 = r 1 ) P ( X 2 = r 2 ) ··· P ( X n = r n ) for any choice of r 1 ,r 2 ,...,r n . Thus, if X 1 , X 2 ,..., X n are mutually independent, then the joint distribution function of the random variable ¯ X = ( X 1 ,X 2 ,...,X n ) is just the product of the individual distribution functions. When two random variables are mutually independent, we shall say more briefly that they are. Bayes’ theorem P ( A | B ) = P ( A ) P ( B | A ) P ( B ) = P ( A ∩ B ) P ( B ) . Remark • Bayes’ theorem is also called the inverse probability law . • It was first used by an 18th century monk named Rev. Thomas Bayes (1702-1761)....
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- Fall '08
- Statistics, Probability theory, Xn