2.1 Example_Class_1_Solution

# 2.1 Example_Class_1_Solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: 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 (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 !...
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2.1 Example_Class_1_Solution - THE UNIVERSITY OF HONG KONG...

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