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ho9-sol - CS237. Problem Set 9: Geometric Distribution,...

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Unformatted text preview: CS237. Problem Set 9: Geometric Distribution, Normal Distribution and Central Limit Theorem Ungraded. Please complete by Thursday April 7. April 7, 2011 Reading. Schaums Chapter 6.1 - 6.6. Extra Practice. Schaums Problems 6.21 - 6.36. Exercise 1. A page contains 30,000 random digits from 0 to 9. Each digit appears with equal probability. 1. What is the expected number of times that the digit 3 appears on the page? Ilir notes: If X denotes the number of 3 , then X : B (30 , 000;1 / 10) , so E [ X ] = 30 , 000 1 / 10 = 3 , 000 . 2. Find the probability that the digit 3 appears more than 10,000 times. Ilir notes: Lets keep the notation from above. We need to find P [ X > 10 , 000] . We have 30 , 000 X i =10 , 001 30 , 000 i ! (1 / 10) i (9 / 10) 30 , 000- i . Using the complement, we have 1- P [ X 10 , 000] . We can approximate this using the normal distribution: 1- NP [ X 10 , 000 . 5] . We know that = 3 , 000 , 2 = 2 , 700 and = 51 . 96 . In standard units, 10 , 000 . 5 corresponds to . 777 . Therefore, we have 1- NP ( Z . 77) = 1- . 2794 = 0 . 7206 . 3. Find the probability that the digit 3 appears between 12,700 and 14,500 times. Ilir notes: Similarly as above, we can approximate the value of P [12 , 700 . 5 X 14 , 499 . 5] using normal distribution. We need to find NP [12 , 700 . 5 X 14 , 499 . 5] . In standard units we have NP [1 . 077 Z 1 . 277] = (1 . 277)- (1 . 077) = 0 . 3980- . 3577 = 0 . 0403 .....
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This note was uploaded on 04/14/2011 for the course CS 237 taught by Professor Goldberg during the Spring '11 term at BU.

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ho9-sol - CS237. Problem Set 9: Geometric Distribution,...

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