Simulation and modeling - Old Questions

Independent test determines whether a random number generator is producing independent random number in a sequence. Test for independence includes the three types of tests as given below:

1) Autocorrelation Test tests the correlation between numbers and compares the sample correlation to the expected correlation of zero.

2) Gap test Counts the number of digits that appear between repetitions of particular digit and then uses the Kolmogorov-Smirnov test to compare with the expected size of gaps,

3) Poker test: Treats numbers grouped together as a poker hand. Then the hands obtained are compared to what is expected using the chi-square test.

Now,

The probabilities associated with each of the given combination is given by

P (four different digits) = ⁴C₄ * (10/10) * (9/10) * (8/10) * (7/10) = 0.504

P (one pair) = ⁴C₂ * (10/10) * (1/10) * (9/10) * (8/10) = 0.432

P (two pair) = (⁴C₂/2)*(10/10) * (1/10) * (9/10) * (1/10) = 0.027

P (three digits of a kind) = ⁴C₃ * (10/10) * (1/10) * (1/10) * (9/10) = 0.036

P (four digits of a kind) = ⁴C_{4 *} (10/10) * (1/10) * (1/10) * (1/10) = 0.001

Now the calculation table for the Chi-square statistics is:

 = 25. 183

and Given  _{α, N} = _{0.05, 4}= 9.49

Here the calculated value of chi-square is 25.183 which is greater than the given tabulated value of chi- square so we reject the null hypothesis of independence between given numbers.