A Random Number Generator Test Suite for the C++ ... - ETH Zürich

n3. Analyzing StatisticsIn this section we describe the χ 2 test and the Kolmogorov-Smirnov test. Both are designedto check if the measured distribution is similar to the expected distribution. So we cancompare different distributions. Later on we describe the gaussian test which is based on thegaussian normal distribution.A detailed description for the outlined C++ classes can be found in the section aboutimplementing additional tests 6.1.3.1. χ 2 test (“Chi-square” test 1 )The χ 2 -Test is perhaps the best known statistical test. It is based on a comparison betweenthe empirical distribution function and the theoretically expected distribution. The empiricaldistribution is based on the results of the random process.The n measured random values must be divided in k classes I 1 I 2contain N 1¤ 2¤ k¥ ¢¡¢¡¢¡£values.k . The classesIN N NFor each class, the expected number of values must be calculated with the expecteddistribution¡¢¡¢¡¤function p i for a given p i (p p§i¨) i¥ N¦Considering the squares of the differences between the measured values and the expectedvalues gives the χ 2 valuei¥ Nkχ ∑i©1§n i 2¥npnp i¥i¨21nk∑i©12 inp i(3.1)With k classes, there are 1 degrees of freedom in the χ 2 distribution. Lookingup for χ 2 and ν in “χ 2 distribution” tables, which can be found in [16], [3], theprobability being above or below the given χ 2 can be found.k ν¥Calculating the probability of a χ 2 value is not such an easy task, but there is analgorithm published by Hill and Pike, which can be used, see [11], [12], [14].Example: Throwing a dieAfter throwing a die 120 times we get the following resultsvalue 1 2 3 4 5 6# observed 15 19 22 21 17 261 Sometimes the “χ 2 test” stands for the Equidistribution test.4