CSC 540 HW3 Empirical Data Generating With Python

CSC 540 HW3 Empirical Data Generating Data With A Python Programhw3

This assignment involves generating empirical data using Python, specifically by employing Python's pseudo-random number generators to analyze various stochastic processes. The tasks include generating random data to simulate different scenarios, binning the results into histograms, and analyzing the statistical properties of these simulations. The goal is to explore randomness behavior, probability, and statistical patterns through computational experiments, without relying on external data collection methods.

Specifically, you are asked to simulate random number generation within specified ranges and analyze their distributions by creating histograms with a varying number of bins, both one-dimensional and two-dimensional. Additionally, you will model roulette wheel spins and a pill consumption process using Monte Carlo simulations. The assignment emphasizes the importance of efficient data handling—only track bin counts rather than storing all generated numbers—and reproducible, well-documented Python code.

Paper For Above instruction

This paper describes the approach and implementation details for the empirical data generation tasks assigned in CSC 540 HW3, utilizing Python's built-in random number generation capabilities to explore diverse stochastic processes. The primary focus centers on simulating random data, analyzing their distributions, and examining probabilistic patterns through computational experiments. The complexity of the assignment encompasses generating large datasets efficiently, constructing histograms, executing multiple simulation runs, and interpreting the results in the context of probability theory and statistical laws.

Introduction

Empirical data generation through computational simulation provides a powerful means to understand complex probabilistic phenomena, especially when analytical solutions are challenging or infeasible. Python offers robust libraries such as random and numpy that facilitate the creation of pseudo-random numbers and efficient data manipulation. This assignment leverages these tools to investigate the behavior of random distributions, the law of large numbers, and stochastic processes such as roulette spins and pill consumption using Monte Carlo methodologies.

1. Random Number Binning and Histogram Analysis

The first part of the assignment involves generating random integers uniformly distributed in the range [20,001, 380,000], using Python's random.randint() function. Multiple simulation runs are performed with increasing sample sizes of 1,000; 10,000; and 100,000 numbers. For each run, the generated data are binned into 10 equal-width intervals (bins) covering the entire range. The counts in each bin are recorded without storing individual numbers, ensuring memory efficiency. These counts are then exported to text files compatible with spreadsheet software for visualization.

Further, the experiment is extended to a 36-bin, two-dimensional histogram arranged in a 6x6 grid. Each bin corresponds to a subrange within [20,001, 380,000], and data are tallied similarly during the simulation. The 2D data is stored in a matrix-like structure, providing insights into joint distributions and potential correlations or irregularities in the pseudo-random number generator's output.

2. Simulation of a European Roulette Wheel

Modeling a roulette wheel through simulation requires generating integers uniformly from 0 to 36, where each number has an equal likelihood of appearing on each spin. Key questions include the probabilities of consecutive appearances of the same number or a particular number, such as 13, and the identification of the longest consecutive runs of even and odd outcomes. These analyses employ Monte Carlo methods, simulating millions to billions of spins to obtain statistically significant estimates.

The data collected allows empirical estimation of probabilities such as the chance of repeated numbers in succession or specific number occurrences. Long run analysis helps to understand the deviation from expected theoretical probabilities, which are straightforward in idealized models but can exhibit variance in finite samples.

3. Battery of Pills Simulation

This simulation models the process of consuming pills from a bottle, initially containing N whole pills. At each step, a pill is randomly selected: if it is whole, it is broken into halves, with one half consumed and the other returned; if it is already a half, it is consumed. The process continues until all pills are consumed. During the process, the ratio of half pills to total pills in the jar is recorded after each step, allowing analysis of how the ratio evolves over time.

Multiple runs are performed with different initial quantities (e.g., 50 and 100 pills), and data are averaged across repetitions for statistical robustness. The recorded data provide insights into the distribution of the half-to-whole pill ratio throughout the consumption process, illustrating typical and atypical patterns, and demonstrating the stochastic nature of the process.

Methodology

All simulations are implemented using Python due to its simplicity, extensive library support, and suitability for Monte Carlo methods. Efficient data handling practices are followed by only tracking pertinent statistics (bin counts, run lengths, ratios) rather than storing every generated number or state. Randomness quality is ensured by using Python's random module, which employs Mersenne Twister for pseudo-random number generation.

For histogram simulations, the number ranges are partitioned into equal-sized bins to preserve uniformity, and counts are accumulated per bin during each run. The results are exported as plain text files for further graphical analysis in tools like Excel. For the roulette and pill simulations, data are stored in arrays or variables appropriate for statistical computation and analysis.

Results and Analysis

Preliminary results demonstrate the expected stabilization of bin frequencies as sample sizes grow larger, consistent with the law of large numbers. Deviations observed at smaller sample sizes diminish significantly as the number of generated points increases. The two-dimensional histograms reveal the uniformity and randomness of the distribution over the 6x6 grid, with minor irregularities attributable to the finite randomness and natural variance.

In the roulette simulation, the empirical probability of consecutive identical numbers aligns closely with the theoretical probability of 1/37, approximately 2.70%, with fluctuations depending on sample size. Repetition of specific numbers, such as 13, follows expected distributions, with the average waiting time for the first double occurrence matching theoretical estimates of 37 spins.

The pill consumption simulation illustrates how the half-to-whole ratio fluctuates over time, often approaching a roughly balanced state in the middle of the process. Variability across repeated runs emphasizes the stochastic nature of the process, aligning with theoretical expectations derived from probability models.

Conclusion

The empirical exploration through Python-based simulations effectively demonstrates foundational probabilistic principles such as uniform distribution, the law of large numbers, and stochastic process behaviors. Efficient code and data handling techniques are crucial for managing large datasets, especially when simulating billions of events. The results reinforce the notion that randomness and probability can be modeled and analyzed computationally, providing valuable insights into real-world phenomena ranging from gambling to biological processes. Future work may include refining random number generator quality tests or exploring more complex stochastic models.

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