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# hw1 - in order to meet the regulation Assume that the...

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Homework 1 (due Jan. 26) 1. Consider the ship insurance business of about 300 years ago. The probability of any ship being lost at sea is 0.015, independent of other ships. Suppose that the goal of every insurance company is to have the probability of a financial loss not exceeding 0.05. A small company insures 1000 ships and a large company insures 2000 ships. Can the large company afford a smaller premium while still achieving the goal of less than 0.05 financial loss probability? How much smaller can the premium be? The payoff in case of a loss is 100,000 for any ship. Hint: find, for each company, the smallest number of ships N so that the event of losing more than N ships has probability of less than 0.05. You can use Excel for that. 2. A pharmaceutical company sells serum in vials with nominal volume of 2.5 ml. A regulation specifies that no more than 1% of all vials should be under-filled. The machine filling the vials does so with a standard deviation of 0.1 ml. How much serum will the company have to put in a vial on average

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Unformatted text preview: in order to meet the regulation? Assume that the volume of serum in a vial is normally distributed. 3. Which of the following functions can represent a probability density functions for some random variable? For such functions, ﬁnd the correspond-ing cumulative distribution functions. a) f ( x ) = if x < if x > 1-1 if 0 < x < 1 b) f ( x ) = 0 if x < 0 if x > 1 1 if 0 < x < 1 c) f ( x ) = if x < if x > 10 / 11 1 . 1 if 0 < x < 10 / 11 d) f ( x ) = if x < if x > 1 1 . 1 if 0 < x < 1 e) f ( x ) = ‰ if x < 5exp(-5 x ) if x > 1 f) f ( x ) = ‰ if x < 5(exp(-5 x )-. 1) if x > g) f ( x ) = 1 √ 2 π exp(-x 2 ) 4. Find the value of C such that f ( x ) is a probability density function for some random variable. a) f ( x ) = ‰ if x < C exp(-λx ) if x > b) f ( x ) = C exp(-λ | x | ) c) f ( x ) = if x < Cx if 0 < x < 3 if x > 3 2...
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