03 - BTRY/STSCI 4080 Homework # 3 Due Date: 9/23/10 Problem...

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Due Date: 9/23/10 Problem numbers beginning with P (e.g., P1) denote problems from the “PROB- LEMS” section; those beginning with T (e.g., T1) denote problems taken from the “THEORETICAL EXERCISES” section. 1. Ross: p 16, P8 2. Ross: p 16, P10 3. Ross: p 16, P15 4. Ross: p 16, P19(c) 5. Ross: p 17, P25 6. Ross: p 17, P33 ( Hint: try to use Prop. 6.1 ) 7. Assume n > 0 is an integer. (a) Using only algebraic manipulations, verify the following identities: i. n k =1 ( 1) k ( n k ) = 1 . ii. n k =0 k ( n k ) = n 2 n - 1 . (See also Ross: p 18, T 12(a)) iii. n k =1 k 2 ( n k ) = n ( n + 1)2 n - 2 . (See also Ross: p 18, T 12(b)) iv. n k =1 ( 1) k +1 k ( n k ) = 0 . Hint: in most cases, you should be able to use the Binomial Theorem. (b) Write a r program that conFrms each of (a)(i) - (a)(iv) holds for n = 1 , 2 , . . . , 25. Hint: the r command choose(n,k) computes ( n k ) . See also factorial and lfactorial . 8.
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03 - BTRY/STSCI 4080 Homework # 3 Due Date: 9/23/10 Problem...

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