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Assignment2-Solutions

Assignment2-Solutions - 1 Assignment 2 Solutions...

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1 Assignment 2- Solutions- STAT220-Fall 2010 Instructor: Kamyar Moshksar Problem 1 - In your course notes, you can find the following identity at the end of section 3 in the “Review of Useful Series and Sums” subsection: min { a,n } X x =max { 0 ,n - b } a x b n - x = a + b n . ( * ) This identity is proved in your course notes using Binomial Expansion. In this problem, you will prove this identity probabilistically. We have a box that contains a + b bulbs where a bulbs are defective and the rest ( b bulbs) are fine. We pick n bulbs at random. (a) what is the probability that exactly x of the n picked bulbs are defective? Answer : ( a x )( b n - x ) ( a + b n ) . In particular, we need that 0 x a and 0 n - x b . This yields max { 0 , n - b } ≤ x min { a, n } . (b) Prove the identity in (*) using the result in part (a). Answer : The sample space that shows the number of defective bulbs is S = { max { 0 , n - b } , max { 0 , n - b } + 1 , max { 0 , n - b } + 2 , · · · , min { a, n } - 1 , min { a, n }} . The sum of the probabilities of these numbers must be 1 , as P( S ) = 1 . Therefore, min { a,n } X x =max { 0 ,n - b } ( a x )( b n - x ) ( a + b n ) = 1 . Multiplying both sides by ( a + b n ) yields the result. Problem 2- We have a coin with the property that P( Head ) = p and P( Tail ) = 1 - p . This coin is tossed repeatedly and independently until a Tail appears for the first time. Find the probability that the number of required tosses is even?
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