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Does increasing the sample size by random number generation change the distribution?

I have a sample of size 8. Each sample value represents the number of bus arrivals at a bus stop every 15 minutes. But I wanted to apply the chi-square test to verify the fitting with the Poisson distribution. So, for every 15 minute interval, I generated 15 random numbers. So I got a new sample size 120.

The numbers were generated following a uniform distribution. See an example:

I had the following sample size 8:

A={8, 13, 13, 14, 15, 11, 16, 11}

My new sample size 120 is:

B={0, 1, 0, 1, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 1, 2, 1, 0, 0, 0, 1, 1, 1, 0, 2, 0, 1, 2, 0, 2, 1, 3, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 1, 2, 2, 0, 2, 1, 1, 0, 0,,1, 1, 1, 1, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 3, 0, 3, 2, 0, 2, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 0}

Notice that the sum of the 1º to 15º is equal to 8, the 16º to the 30º equals 13 and so on. I would like to know if the distribution of A sample always will be equal to the distribution of B sample, for any random sample that I generate this way.

How much does this changes the characteristics of the system?

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    $\begingroup$ Exactly how do you suppose that generating random values is adding any information to your data? $\endgroup$ – whuber Jun 10 at 20:26

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