This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CS 70 Discrete Mathematics and Probability Theory Fall 2010 Tse/Wagner Soln 10 1. (50 pts.) Expectations Solve each of the following problems using linearity of expectation. Clearly explain your methods. (Hint: for each problem, think about what the appropriate random variables should be and define them explicitly.) (a) A monkey types at a 26letter keyboard with one key corresponding to each of the lowercase English letters. Each keystroke is chosen independently and uniformly at random from the 26 possibilities. If the monkey types 1 million letters, what is the expected number of times the sequence bonbon" appears? Answer: Define the random variable X to be the number of times bonbon appears in the 1 million letters the monkey types. We express X as the sum X = 10 6 5 i = 1 X i where the random variable X i is X i = ( 1 if characters i ,..., i + 5 are bonbon otherwise. Since each letter is independent, the probability that any given string of 6 letters is bonbon is 1 26 6 , and therefore Pr [ X i = 1 ] = 1 26 6 . Thus, E ( X i ) = 1 26 6 and so by linearity of expectation E ( X ) = E ( 10 6 5 i = 1 X i ) = 10 6 5 i = 1 E ( X i ) = 10 6 5 i = 1 1 26 6 = ( 10 6 5 ) 1 26 6 . Comment: This solution is fine, despite the fact that the X i s are not mutually independent. Linearity of expectation can be applied even if the underlying random variables are not independent. (b) A coin with Heads probability p is flipped n times. A run is a maximal sequence of consecutive flips that are all the same. (Thus, for example, the sequence HTHHHTTH with n = 8 has five runs.) Show that the expected number of runs is 1 + 2 ( n 1 ) p ( 1 p ) . Justify your answer carefully. Answer: Again, we use linearity of expectation. Define the random variable X to be the number of runs in a sequence, and define the indicator random variable X i as X i = ( 1 if the i th flip is the beginning of a run otherwise....
View
Full
Document
This note was uploaded on 12/08/2010 for the course CS 70 taught by Professor Papadimitrou during the Fall '08 term at University of California, Berkeley.
 Fall '08
 PAPADIMITROU

Click to edit the document details