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# PROBABILITY FOR ENGR APPL ECSE 4500

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This 2 page Class Notes was uploaded by Immanuel Brakus PhD on Monday October 19, 2015. The Class Notes belongs to ECSE 4500 at Rensselaer Polytechnic Institute taught by John Woods in Fall. Since its upload, it has received 68 views. For similar materials see /class/224760/ecse-4500-rensselaer-polytechnic-institute in ELECTRICAL AND COMPUTER ENGINEERING at Rensselaer Polytechnic Institute.

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Date Created: 10/19/15

Rensselaer Electrical Computer and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 1 Wednesday August 27 2008 Review of Relevant Mathematics ref Appendix A in Stark and Woods SampW text 1 sum of Geometric Series and related a Consider the formula 1 oo 2a 1 n0 7 1 Under what conditions on a is the formula correct Why b More generally7 we have7 for n2 gt n1 n2 n1 n11 17a 7 nn1 which holds for any value of 1 except a 1 Find this result by using the formula as 5 ani1 7 am where S Zn a Why is the above formula true for the geometric series nn1 c Now de ne a generating function as oo Zanz 710 1 Show that G1 20012 whenever oil lt 1 2 Show that G 1 2200 not 17am whenever lal lt 1 This result will be used for nding the average value of the soecalled geometric random variable 3 How can we nd 2200 n2a 2 Integration by Parts The integration by parts formula is as follows uxdvx Z 7 abvxduac a Use the integral Abdomen to derive the above formula b Use integration by parts to perform the following integral 5 xexdac 0 c Perform the following integral om 3 exp72x 4dac d What is the value of the integral oo 12 exp721dac 3 di erention of integrals We Will have need to differentiate integrals7 Where the upper limit is a variable or function d x 2d dag0 11 d 0 2d dag1111 as E 0 a Evaluate the derivative b Evaluate the derivative c What is the derivative 302 y2dy 4 Mathematical Induction The principle of mathematical induction is stated as Let S be a set of integers lf 1 E S and k E 8 implies that 16 1 E S then all positive integers are in 8 Use this principle to prove7 for 0 lt a lt 117 that a gt If for all positive integers n

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