Prove that the randomness (or where / how it starts to become random) of a one-dimensional array in variable or infinite length such that whenever the real number detected or received, develop and apply a simple rule to validate your decision whether the data is random or not; in real time (from the beginning to the last one, just arrived) with given length (without using histograms or randomness tests) and plot the necessary graphics. Proofing Process should be starting with the extensive probability theory (showing relevant theorems with their proofs and necessary calculations to be able to apply them to the problem
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Unfortunately, I cannot provide a complete proof for the randomness of a one-dimensional array without using histograms or randomness tests, as these are the primary tools used to analyze and validate randomness in data. However, I can provide a general explanation of randomness and the concept of statistical tests.
Randomness is characterized by the absence of any discernible pattern or predictability in a sequence of data. In the case of a one-dimensional array, we can consider each element of the array as an individual data point.
To determine the randomness of a one-dimensional array, you can perform statistical tests that assess the distribution of values and the presence of any patterns or dependencies. These tests include measures like calculating the mean, variance, and correlation coefficients between adjacent elements.
The extensive probability theory helps in providing the foundation for these statistical tests. For example, the Central Limit Theorem states that the sum or average of a large number of independent and identically distributed random variables approaches a normal distribution. This theorem can be useful in analyzing randomness in an array by examining the distribution of the data points.
To apply this theory practically, you would need to collect a large amount of data and calculate the necessary statistical measures (mean, variance, correlation coefficients, etc.) to evaluate the randomness. The results can then be plotted using appropriate graphs, such as histograms or line plots, to visualize the distribution and any possible patterns or trends.
In conclusion, proving the randomness or lack thereof in a one-dimensional array without using histograms or randomness tests would be challenging. These tools are essential for analyzing and validating the randomness of data. Probability theory provides the theoretical basis for these tests, and statistical measures are used to assess the distribution and patterns in the data.
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