Discussion4

Discussion4 - Discussion 4 Emily Mower Topics to be covered...

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Unformatted text preview: Discussion 4 Emily Mower February 17, 2010 Topics to be covered: • Survivors guide: pages 6-8 • Description of mixed random variables • Software error vs. maintenance costs • Textbook problem 2.100 1 1 Survivor’s Guide 1.1 Inequalities and Their Graphs The best way to think about how to deal with inequalities is to figure out where the function is zero. Then plug in values on either side of this line to figure out which side to shade. See Figures 1 and 2 for examples. Figure 1: An example of the graphical depiction of inequalities . Figure 2: An example of the graphical depiction of inequalities . 2 1.2 Series Expansions Let’s start with the finite series: a + ar + ar 2 + ... + ar n- 1 • Let s = 1 + r + r 2 + ... + r n- 1 • Thus, rs = r + r 2 + r 3 + ... + r n • Subtract rs from s : s- rs = 1- r n • Rearrange to solve for s: s = 1- r n 1- r • Multiply by a factor, a: as = a- ar n 1- r Thus, for the finite series: a + ar + ar 2 + ... + ar n- 1 we can say: a + ar + ar 2 + ... + ar n- 1 = a- ar n 1- r When a = 1, this simplifies to: 1 + r + r 2 + ... + r n- 1 = 1- r n 1- r Let’s now look at the infinite series: a + ar + ar 2 + ... If | r | < 1 , then as n → ∞ : a + ar + ar 2 + ... = a 1- r , | r | < 1 When a = 1, this simplifies to: 1 + r + r 2 + ... = 1 1- r , | r | < 1 Let’s differentiate the infinite series: 1 + r + r 2 + r 3 + ... D (1 + r + r 2 + r 3 + ... ) = 1 + 2 r + 3 r 2 + r 3 + ... = D ( 1 1- r ) = 1 (1- r ) 2 , | r | < 1 3 Let’s differentiate the finite series: 1 + r + r 2 + r 3 + ... + r n D (1 + r + r 2 + r 3 + ... + r n- 1 ) = 1 + 2 r + 3 r 2 + r 3 + ... + nr n- 2 = D ( 1- x n 1- x ) = nr n- ( n + 1) r n- 1 + 1 (1- r ) 2 Now let’s look at e . In 1683, Johann Bernoulli defined e as: e = lim n →∞ (1 + 1 n ) n e x = lim n →∞ (1 + x n ) n If we want develop an expansion of the e or e x we need to use the binomial theorem....
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Discussion4 - Discussion 4 Emily Mower Topics to be covered...

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