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exampleclass1solution - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 1 Review Combinatorial Analysis (a) Multiplication Principle (b) Selection of r from n distinct objects: With replacement Without replacement Ordered n r n P r = n ! ( n - r )! Unordered ( n + r - 1 r ) = ( n + r - 1)! r ! × ( n - 1)! ( n r ) = n ! r ! × ( n - r )! (c) Arrangement of n objects, with r distinct types. In other words, there are n 1 objects of type 1, n 2 objects of type 2, ..., n r objects of type r . (e.g. number of different letter arrangements can be formed using the letters STATISTICS.) n ! n 1 ! × n 2 ! × · · · × n r ! (d) Partition of n distinct objects into r distinct groups with specified size n 1 , n 2 , · · · , n r . (e.g. number of ways in dividing a class of 40 into groups of 10, 10 and 20.) n n 1 , n 2 , · · · , n r = n ! n 1 ! × n 2 ! × · · · × n r ! (e) Partition of n indistinguishable objects into r distinct groups (i.e. we only concern the number of objects in each group). n + r - 1 n = ( n + r - 1)! n ! × ( r - 1)! Set Theory and Mathematical Theory of Probability (a) De Morgan’s law n [ i =1 E i ! c = n \ i =1 E c i , n \ i =1 E i ! c = n [ i =1 E c i , where n can also be . 1
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(b) Language of Probability (i) Mutually exclusive A 1 , A 2 , · · · , A n are mutually exclusive if A i A j = φ for all i 6 = j . (ii) Exhaustive A 1 , A 2 , · · · , A n are exhaustive if A 1 A 2 ∪ · · · ∪ A n = Ω. (iii) Partition A 1 , A 2 , · · · , A n is called a partition if the events are mutually exclusive and exhaustive. (iv) Complement The complement of event A is the collection of outcomes not in A, i.e. A c = Ω \ A (c) Kolmogorov’s Axiom (i) P ( A ) 0 for any event A (ii) P (Ω) = 1 (iii) For any sequence of mutually exclusive events A 1 , A 2 , ... , P [ i =1 A i ! = X i =1 P ( A i ) (Countable additivity) Mnemonic trick: length of the union of disjoint segments is equal to the sum of their individual lengths.
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