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A bimodal distribution as shown in Fig 6-1 can best be decomposed by tting two normal distributions to the data Four parameters have to be determined: mean1 , 1 , mean2 , and 2 If the modes are well separated, the means can be estimated at each node and tting of two normal curves will be simple It helps if the s are equal or have a predetermined relationship Sometimes the position of the means can be determined by alternative reasoning Even when the shape of the distribution does not match any known distribution, there is still a conservative statistical tool available, Chebyshev s inequality theorem, to describe the behavior of the variables Given the mean and the standard deviation, which are always computable for a set of observations, Chebyshev s inequality provides directly the probability of values from the set of observations xi , i = 1, , n falling outside a range The range, rg, is de ned in terms of a number, c, of standard deviations, , to either side of the mean of x The mean and standard deviation are computed from the observations without any assumptions regarding the behavior of the distribution If the sample observations are not biased, then for some k
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p(xk not in rg)
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mean c < rg < mean + c
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This relationship can, for instance, be used to state that given any observed distri bution, 90% of the values will be within 10 = 3162 to either side of its mean If the distribution is approximately symmetrical, we can assume that only 5% of the values exceed the mean + 3162 The worst case is that all 10% of the values exceed the mean + 3162 If the distribution has a single mode and is symmetric, then p(x not in rg) 2 3c2 6-23
and, correspondingly, the probability that a value, xk , of the observations exceeds the mean by c is 1 p(xk > mean + c ) 2 6-24 3c
Example 6-8
Estmating le over ow with the Chebyshef inequality
For the direct le used to illustrate Eq 6-20, the Chebyshev inequality can be used to evaluate the chance of exceeding a primary over ow area of a given size The distribution of over ows is not symmetric, so that the worst case approximation should be used Given was o = 265, = 258, and we wish to guarantee that in all but 1% of the cases the records t into the primary over ow area Here and the required space o = o + 10 = 523 c = 100 = 10 This value is based on the sample observations Multiple samples should be used to verify stability of the result
Sec 6-2 6-1-7 Other Statistics
Simulation
At times, an observed response is a function of multiple variables With assumption of independence among these variables we can separate the e ects given a su ciently large number of observations Two types of analysis methods are in use for this work: multiple regression and analysis of variance Regression analysis determines a line describing a linear relationship between the individual variables and the resulting dependent variable Many relationships in computer systems are not linear Analysis of variance relates causes and e ects by discrete groupings and has been used to analyze the performance of paging operating systems based on the variables: memory size, program size, loading sequence, and paging algorithms Both regression and analysis of variance assume linear relationships of the combination of independent variables on the dependent variable 6-1-8 Help with Statistics It is not necessary to be a statistical expert to use the methods and results that statistical research and development have provided Many statistical packages exist to help perform these calculations Graphical output can make the models being used accessible to people who are not comfortable with the numerical aspects of statistics You cannot expect that statistical experts will solve systems problems The understanding and modeling of the system being evaluated remains as important as the application of analytical techniques This also means that expertise about the computer system being evaluated and its applications must be available as well when an analysis is being made Especially, the actual design process cannot be delegated to specialists in a single narrow area, and neither programmers nor statisticians can be expected to provide acceptable system evaluations by themselves