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### 9_21_11

Course: IE 230, Winter 2010
School: Purdue
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Discrete Variance, Uniform Distribution, Permutations and Combinations Instructor: Andrew Liu September 21, 2011 Textbook sections: 3-4, 3-5, 2-1.4 Review Variance: 2-dice-throw example Probability Mass Function 7/36 Probability 1/6 5/36 1/9 1/12 1/18 1/36 0 2 3 4 5 6 7 8 9 10 11 12 Range of Random Variable E (X ) = 2 1 1 1 1 +3 +4 + + 12 = 7 (make sense?). 36 18 12 36 Textbook sections: 3-4,...

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Discrete Variance, Uniform Distribution, Permutations and Combinations Instructor: Andrew Liu September 21, 2011 Textbook sections: 3-4, 3-5, 2-1.4 Review Variance: 2-dice-throw example Probability Mass Function 7/36 Probability 1/6 5/36 1/9 1/12 1/18 1/36 0 2 3 4 5 6 7 8 9 10 11 12 Range of Random Variable E (X ) = 2 1 1 1 1 +3 +4 + + 12 = 7 (make sense?). 36 18 12 36 Textbook sections: 3-4, 3-5, 2-1.4 Review Variance: 2-dice-throw example Probability Mass Function 7/36 Probability 1/6 5/36 1/9 1/12 1/18 1/36 0 2 3 4 5 6 7 8 9 10 11 12 Range of Random Variable E (X ) = 2 1 1 1 1 +3 +4 + + 12 = 7 (make sense?). 36 18 12 36 V (X ) = E (X 2 ) [E (X )]2 = 1 1 1 22 + 32 + + 122 36 18 36 72 5.83. Textbook sections: 3-4, 3-5, 2-1.4 Review Variance: 2-dice-throw example Probability Mass Function 7/36 Probability 1/6 5/36 1/9 1/12 1/18 1/36 0 2 3 4 5 6 7 8 9 10 11 12 Range of Random Variable E (X ) = 2 1 1 1 1 +3 +4 + + 12 = 7 (make sense?). 36 18 12 36 V (X ) = E (X 2 ) [E (X )]2 = 1 1 1 22 + 32 + + 122 36 18 36 72 5.83. (X ) 2.42. Textbook sections: 3-4, 3-5, 2-1.4 Examples (cont.) Given a random variable X . Suppose its probability mass function is given as follows. 81 f (x ) = ( )( )x , x = 1, 2, 3. 72 Calculate V (X ). Textbook sections: 3-4, 3-5, 2-1.4 Examples (cont.) Given a random variable X . Suppose its probability mass function is given as follows. 81 f (x ) = ( )( )x , x = 1, 2, 3. 72 Calculate V (X ). Solution: (Just use the formula of variance!) E (X ) = X x 81 81 81 xf (x ) = 1 ( )( ) + 2 ( )( )2 + 3 ( )( )3 1.57. 72 72 72 V (X ) = E (X 2 ) [E (X )]2 = Textbook sections: 3-4, 3-5, 2-1.4 Examples (cont.) Given a random variable X . Suppose its probability mass function is given as follows. 81 f (x ) = ( )( )x , x = 1, 2, 3. 72 Calculate V (X ). Solution: (Just use the formula of variance!) E (X ) = X x 81 81 81 xf (x ) = 1 ( )( ) + 2 ( )( )2 + 3 ( )( )3 1.57. 72 72 72 V (X ) = E (X 2 ) [E (X )]2 = 12 f (1) + 22 f (2) + 32 f (3) 1.572 1.43. Textbook sections: 3-4, 3-5, 2-1.4 Discrete Uniform Distribution Suppose that the range of X is {x1 , x2 , . . . , xn }. X has a discrete uniform distribution if 1 f (xi ) = . n Textbook sections: 3-4, 3-5, 2-1.4 Discrete Uniform Distribution Suppose that the range of X is {x1 , x2 , . . . , xn }. X has a discrete uniform distribution if 1 f (xi ) = . n Example: the face value of a fair die throw. Textbook sections: 3-4, 3-5, 2-1.4 Discrete Uniform Distribution Suppose that the range of X is {x1 , x2 , . . . , xn }. X has a discrete uniform distribution if 1 f (xi ) = . n Example: the face value of a fair die throw. Q: Is the sum of the face values of two fair dice throw a discrete uniform random variable? Textbook sections: 3-4, 3-5, 2-1.4 Discrete Uniform Distribution (cont.) Mean and Variance of Discrete Uniform Distribution Suppose X is a discrete uniform random variable on consecutive integers {a, a + 1, a + 2, . . . , b }. Then E (X ) = b+a (b a + 1)2 1 , V (X ) = . 2 12 Textbook sections: 3-4, 3-5, 2-1.4 Discrete Uniform Distribution (cont.) Mean and Variance of Discrete Uniform Distribution Suppose X is a discrete uniform random variable on consecutive integers {a, a + 1, a + 2, . . . , b }. Then E (X ) = b+a (b a + 1)2 1 , V (X ) = . 2 12 Examples E.g. 1. Let X denote the face value of a fair die throw. Calculate E (X ) and V (X ). Textbook sections: 3-4, 3-5, 2-1.4 Discrete Uniform Distribution (cont.) Mean and Variance of Discrete Uniform Distribution Suppose X is a discrete uniform random variable on consecutive integers {a, a + 1, a + 2, . . . , b }. Then E (X ) = b+a (b a + 1)2 1 , V (X ) = . 2 12 Examples E.g. 1. Let X denote the face value of a fair die throw. Calculate E (X ) and V (X ). E.g. 2 Let the random variable X have a discrete uniform distribution on the integers 1 x 3. E (X ) =?, V (X ) =? Textbook sections: 3-4, 3-5, 2-1.4 Binomial Distribution To introduce binomial distribution, we need to learn counting techniques rst. Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Permutations Example: White Collar Suppose that Neal and Peter are trying to open a safe. They know that the code to open the safe contains distinct 4 digits from 0 to 9. If Neal is to exhaust all the possible combinations, how many times does Neal have to try? Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Permutations Example: White Collar Suppose that Neal and Peter are trying to open a safe. They know that the code to open the safe contains 4 distinct digits from 0 to 9. If Neal is to exhaust all the possible combinations, how many times does Neal have to try? Q: are 1234 and 4321 the same codes? A more general question: does order matters? Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Permutations Example: White Collar Suppose that Neal and Peter are trying to open a safe. They know that the code to open the safe contains 4 distinct digits from 0 to 9. If Neal is to exhaust all the possible combinations, how many times does Neal have to try? Q: are 1234 and 4321 the same codes? A more general question: 10 does order matters? (A: 10 * 9 * 8 * 7, denoted as P4 ) Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Combinations E.g. 1. How many possible ways are there to choose 3 items out 4 distinct items? Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Combinations E.g. 1. How many possible ways are there to choose 3 items out 4 distinct items? Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Combinations E.g. 1. How many possible ways are there to choose 3 items out 4 distinct items? E.g.2. Suppose that a department has 10 faculty members. You need to choose 4 faculty members to form a committee to guide your study. How many dierent committees can you form? Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Combinations E.g. 1. How many possible ways are there to choose 3 items out 4 distinct items? E.g.2. Suppose that a department has 10 faculty members. You need to choose 4 faculty members to form a committee to guide your study. How many dierent committees can you form? Q: Is a committee consisting of Steve, Joe, Susan and Kay dierent than that consisting of Kay, Susan, Joe and Steve? More general question: does order matters? Textbook sections: 3-4, 3-5, 2-1.4 Sec 2-1.4 Counting Techniques: Combinations E.g. 1. How many possible ways are there to choose 3 items out 4 distinct items? E.g.2. Suppose that a department has 10 faculty members. You need to choose 4 faculty members to form a committee to guide your study. How many dierent committees can you form? Q: Is a committee consisting of Steve, Joe, Susan and Kay dierent than that consisting of Kay, Susan, Joe and Steve? 10 More general question: does order matters? (A: P4 /4! = 210) Textbook sections: 3-4, 3-5, 2-1.4 Permutations and Combinations Permutation Select r distinct elements from n distinct elements, and order matters! Prn = n (n 1) (n 2) (n r + 1) = n! . (n r )! Textbook sections: 3-4, 3-5, 2-1.4 Permutations and Combinations Permutation Select r distinct elements from n distinct elements, and order matters! Prn = n (n 1) (n 2) (n r + 1) = n! . (n r )! Combinations Select r distinct elements from n distinct elements, and order DOES NOT matter! Crn = n r Pn = r= r! n! (nr )! r! = n! . (n r )!r ! Textbook sections: 3-4, 3-5, 2-1.4 Permutations and Combinations: More Examples E.g. 1 An encyclopedia has eight volumes. In how many ways can the eight volumes be replaced on the shelf? Textbook sections: 3-4, 3-5, 2-1.4 Permutations and Combinations: More Examples E.g. 1 An encyclopedia has eight volumes. In how many ways can the eight volumes be replaced on the shelf? E.g. 2 Mr. Smith is the Chairman of a committee. In how many ways can a committee of 5 be chosen from 10 people (including Mr. Smith) given that Mr. Smith must be one of them? Textbook sections: 3-4, 3-5, 2-1.4 Permutations and Combinations: More Examples E.g. 1 An encyclopedia has eight volumes. In how many ways can the eight volumes be replaced on the shelf? E.g. 2 Mr. Smith is the Chairman of a committee. In how many ways can a committee of 5 be chosen from 10 people (including Mr. Smith) given that Mr. Smith must be one of them? E.g. 3 Picking a President, Vice President, an accountant, and an consultant from a group of 10 people. Textbook sections: 3-4, 3-5, 2-1.4
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