# lect12 - Quantiles Departures From Normality Outline...

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Quantiles Departures From Normality Outline Quantiles Value at Risk (VaR) Departures From Normality Normal Quantile Plot Skewness and Kurtosis 1 / 14 ISOM 2500 Lect 12: Normal Model II

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Quantiles Departures From Normality Value at Risk (VaR) Example: packaging Suppose a box of cereal lists the weight as 16 oz. The boxes are ﬁlled by an automated packaging system, which ﬁlls boxes such that the weights are normally distributed with μ = 16 . 3 oz and σ = 0 . 2 oz. What’s the chance that the weight is exactly 16 oz, as the package label states? What’s the chance that the weight is less than 16 oz? Let X be the weight of cereal in a box. Then X N ( 16 . 3 , 0 . 2 2 ) . We want to compute P ( X < 16 ) = 0 . 067 . 2 / 14 ISOM 2500 Lect 12: Normal Model II
Quantiles Departures From Normality Value at Risk (VaR) Example: packaging Now suppose we want to decrease the chance of an underweight box, to reduce customer complaints. And suppose we can change the mean but not the SD of the packaging system. Where should we set the mean if we want that the chance of an underweight box is only 0.005? Let Y be the weight of cereal in a box ﬁlled by the system with a new mean μ 1 . Then Y N ( μ 1 , 0 . 2 2 ) . We want to ﬁnd a μ 1 such that 0 . 005 = P ( Y < 16 ) = P ± Y - μ 1 0 . 2 < 16 - μ 1 0 . 2 ² = P ± Z < 16 - μ 1 0 . 2 ² . To ﬁnd μ 1 , we need the concept of quantile . .. 3 / 14 ISOM 2500 Lect 12: Normal Model II

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Departures From Normality Value at Risk (VaR) Quantile For any random variable X , its k % quantile or k th percentile is the value x such that P ( X x ) = k % . In the previous example, we want to ﬁnd a value “?” such that P ( Z < ? ) = 0 . 005 , i.e., we want to ﬁnd the 0.5% quantile of the Standard Normal . 4 / 14
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## This note was uploaded on 12/20/2011 for the course ACCT/MGMT 2010 taught by Professor A during the Spring '11 term at HKUST.

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lect12 - Quantiles Departures From Normality Outline...

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