Natural Test for Random Numbers Generator Based on Exponential Distribution

We will prove that when uniformly distributed random numbers are sorted by value, their successive differences are a exponentially distributed random variable Ex(lambda). For a set of n random numbers, the parameters of mathematical expectation and standard deviation is lambda = n(-1). The theorem was verified on four series of 200 sets of 101 random numbers each. The first series was obtained on the basis of decimals of the constant e = 2.718281 ..., the second on the decimals of the constant pi = 3.141592 ..., the third on a Pseudo Random Number generated from Excel function RAND, and the fourth series of True Random Number generated from atmospheric noise. The obtained results confirm the application of the derived theorem in practice.