HW1Sol

HW1Sol - Homework 1 solution 1. Either company needs to...

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Homework 1 – solution 1. Either company needs to adjust its premium so that it can make payoffs for up to x * ships without losing money, where x * is the smallest number such that P ( X > x * ) < 0 . 05 (where X Bin ( n, 0 . 015) is the number of lost ships). Using excel, it’s easy to find that, for the smaller company ( n = 1000), x * = 22, and for the larger company ( n = 2000), x * = 39. So the minimum premium that allows to satisfy the requirement of risk being no more than 0.05 is: 22 · 100000 1000 = 2200, for the smaller company; and 39 · 100000 2000 = 1950, for the larger company. 2. The lifetime of any given bulb is T N (10000 , 1000). a) P ( T > 12000) = 1 - F (12000) = 1 - Φ 12000 - 10000 1000 · = 1 - Φ(2) = 1 - 0 . 977 = 0 . 023. b) P ( T < 9000) = F (9000) = Φ 9000 - 10000 1000 · = Φ( - 1) = 0 . 159. c) P (replacing at least one bulb) = 1 - P (replacing no bulbs) = 1 - P (all T i > 5000) = 1 - P ( T > 5000) 100 = 1 - (1 - F (5000)) 100 = 1 - (1 - Φ( - 5)) 100 = 0 (up to three decimal points). d)
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HW1Sol - Homework 1 solution 1. Either company needs to...

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