These are some of the tests used for testing randomness of a random number generator
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Kolmogrov Smirnov test It is a goodness of fit test and is used for testing whether the random numbers generated from random number generator follow uniform distribution or not
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Runs test It is used to test the independence of random numbers
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Poker test It is also used to test the independence of random numbers
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Spectral test It is a graphical test and is used to test the k-dimensional uniformity of random numbers
Here, the various tests mentioned above are performed on linear congruential generators, RANDU generator and also on randoms numbers generated from R