Poisson distribution with grouping of low frequencies
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/* Poisson Distribution with Grouping of Low Frequencies 6-9 */ DECLARE expfr(0:20); expfr(0) = n*EXP(-mean); left = n-expfr(0); /* number of blocks for assignment */ prod = expfr(0)*mean; DO ov = 1 BY 1 WHILE(prod<ov); /* ok if next expfr > 1 */ expfr(ov) = prod/ov; /* as computed now */ left = left-expfr(ov); prod = expfr(ov)*mean; END; /* used later in Table 6-10 */ last category = ov; expfr(last category) = left;
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Testing for Goodness of Fit
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A more formal test to determine if a distribution ts the data can be made using the chi-square function The chi-square, or 2 , test compares categories of observations and their expectations, but each category should contain at least one expected sample To avoid invalid categories, the frequencies of the Poisson distribution that are less than 1 can be grouped with the last good category as shown in the program of Table 6-9 and Example 6-6
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Poisson distribution of insertions for an indexed-sequential le
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There were 550 insertions into a dense indexed-sequential le of 400 blocks These insertions caused 0, 1, 2, over ow records to be written for each of the blocks The mean number of insertions per block is mean = 550/400 = 138 The expected frequency is computed for n = 400
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No of over ows ov 0 1 2 3 4 5 6 7
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Observations Frequency Records fr(ov) ov fr(ov) 101 138 98 45 11 5 2 0 0 400
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Expectations Frequency expfr(ov) 1014 1391 956 438 151 41 09 02 400
138 196 135 44 25 12 0 0 550
The observed values, listed in Example 6-5 are best inspected again visually A better presentation for that purpose would be a histogram fx as can be generated by the program segment in Table 6-5 The cells of low expected value are combined into a single, namely the last category, by accumulating them in a program loop as shown in Table 6-10 Then the chi-square values are computed, as shown in Table 6-11, to see how well the actual observations t the assumed Poisson distribution The nal results are seen together in Example 6-6
Combining tail values of an observed frequency histogram
/* Combine tail values of observed frequency histogram */ /* 6-10 */ DO i = last category+1 TO number of categories; fx(last category) = fx(last category) + fx(i); END;
Statistical Methods Computation of Chi-square value for goodness of t test
/* Computation of Chi-Square Value for Goodness of Fit */ /* 6-11 */ chisquare = 0; DO ov = 0 TO last category; dif = fx(ov)- expfr(ov); chisqterm = dif**2 / expfr(ov); chisquare = chisquare + chisqterm; PUT DATA( ov, fx(ov), expfr(ov), dif, chisqterm); END; PUT DATA( SUM(fx), SUM(expfr), chisquare); Values obtained for 2 can be compared with standard values, which are based on the assumption that the di erence of distributions was caused by random events These standard values for 2 can be computed as needed using approximations of binomial distributions or can be found in statistical tables and graphs Figure 6-6 presents the standard 2 distribution in graphical form In order to use the 2 distribution, the number of degrees of freedom df has to be known When we distribute our samples over a speci c number of categories c the value of df will be equal to c 1
Testing a Poisson distribution fx(ov)
101 138 98 45 11 7 400
0 1 2 3 4 last n=
1011 1391 956 438 151 53 400
01 11 24 12 41 17 0001 0009 0060 0033 1113 0545 2 = 1761
The value for 2 is 1761 at a df = 5 for the indexed-sequential le observations shown in Example 6-5 The value for this comparison falls within the area of Example 6-6, which is appropriate for most cases which match an expected distribution The point is o -center, close to the good side, so that it seems likely that the le updates are not quite random, but somewhat uniform Perhaps many of the insertions are due to some regular customer activity
A very high value of 2 makes it unlikely that the frequencies are related; a very low value could cause suspicion that the data is arranged to show a beautiful t The chi-square test is useful when distributions are being compared Other tests, such as the t-test and F-test, can be used to compare means and standard deviations obtained from samples with their expected values, if the distribution is known or assumed to be known