ST2131 ONLINE NOTES.pdf - Revision notes ST2131 Ma Hongqiang Contents 1 Combinatorial Analysis 2 2 Axioms of Probability 3 3 Conditional Probability and

ST2131 ONLINE NOTES.pdf - Revision notes ST2131 Ma...

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Revision notes - ST2131 Ma Hongqiang April 18, 2017 Contents 1 Combinatorial Analysis 2 2 Axioms of Probability 3 3 Conditional Probability and Independence 5 4 Random Variables 7 5 Continuous Random Variable 12 6 Jointly Distributed Random Variables 18 7 Properties of Expectation 24 8 Limit Theorems 29 9 Problems 30 1
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1 Combinatorial Analysis Theorem 1.1 (Generalised Basic Principle of Counting) . Suppose that r experiments are to be preformed. If experiment 1 can result in n 1 possible outcomes; experiment 2 can result in n 2 possible outcomes; • · · · experiment r can result in n r possible outcomes; then together there are n 1 n 2 · · · n r possible outcomes of the r experiments. 1.1 Permutations Theorem 1.2 (Permutation of distinct objects) . Suppose there are n distinct objects, then the total number of permutations is n ! . Theorem 1.3 (General principle of permutation) . For n objects of which n 1 are alike, n 2 are alike, . . . , n r are alike, there are n ! n 1 ! n 2 ! · · · n r ! different permutations of the n objects. 1.2 Combinations Theorem 1.4 (General principle of combination) . If there are n distinct objects, of which we choose a group of r items, number of combinations equals n r = n ! r !( n - r )! 1.2.1 Useful Combinatorial Identities 1. For 1 r n , ( n r ) = ( n - 1 r - 1 ) + ( n - 1 r ) 2. ( Binomial Theorem ) Let n be a non-negative integer, then ( x + y ) n = n X k =0 n k x k y n - k 3. n k =0 ( n k ) = 2 n 4. n k =0 ( - 1) k ( n k ) = 0 2
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1.3 Multinomial Coefficients If n 1 + n 2 + · · · + n r = n , we define ( n n 1 ,n 2 , ··· ,n r ) by n n 1 , n 2 , · · · , n r = n ! n 1 ! n 2 ! · · · n r ! Thus ( n n 1 ,n 2 , ··· ,n r ) represents the number of possible divisions of n distinct objects into r distinct groups of respective sizes n 1 , n 2 , · · · , n r . Theorem 1.5 (Multinomial Theorem) . ( x 1 + x 2 + · · · + x r ) n = X ( n 1 ,...,n r ): n 1 + ··· + n r = n n n 1 , n 2 , · · · , n r x n 1 1 x n 2 2 · · · x n r r 1.4 Number of Integer Solutions of Equations Theorem 1.6. There are ( n - 1 r - 1 ) distinct positive integer-valued vectors ( x 1 , x 2 , . . . , x r ) that satisfies the equation x 1 + x 2 + · · · + x r = n where x i > 0 for i = 1 , . . . , r Theorem 1.7. There are ( n + r - 1 r - 1 ) distinct non-negative integer-valued vectors ( x 1 , x 2 , . . . , x r ) that satisfies the equation x 1 + x 2 + · · · + x r = n where x i > 0 for i = 1 , . . . , r 2 Axioms of Probability 2.1 Sample Space and Events The basic objects of probability is an experiment : an activity or procedure that produces distinct, well-defined possibilities called outcomes . The sample space is the set of all possible outcomes of an experiment, usually denoted by S . Any subset E of the sample space is an event . 2.2 Axions of probability Probability , denoted by P , is a function on the collection of events satisfying (i) For any event A , 0 P ( A ) 1 (ii) Let S be the sample space, then P ( S ) = 1 3
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(iii) For any sequence of mutually exclusive events A 1 , A 2 , . . . (that is A i A j = when i 6 = j ) P ( i =1 A i ) = X i =1 P ( A i ) 2.3 Properties of Probability Theorem 2.1. P ( ) = 0.
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