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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.
 Spring '11
 Goldberg

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