CSC 540 Individual Programming Assignment HW3 Your Program S

Csc 540 Individual Programming Assignment Hw3your Program Should Use

Csc 540 Individual Programming Assignment Hw3your Program Should Use

CSC 540 INDIVIDUAL PROGRAMMING ASSIGNMENT, HW3 Your program should use the Free Pascal compiler described in CSC 540. When you hand in your assignment to the digital dropbox, it should be a .PAS file, not zipped. Your assignment is to explore the behavior of your Pascal random number generator by throwing random numbers between 20001 and 380000 (inclusive). At first you will bin the results into 10 equal histogram bins. After each simulation run, output the histogram results into a text file. Then import the text file into Excel (or some other graphing program) and use a bar graph to display your histogram results. You should run at least three different simulations to generate numbers for your histogram bins, with the option to do more. Feel free to run each more than once to observe variability, utilizing the randomize function in Free Pascal to see if different runs produce different outcomes.

You will conduct simulations as follows:

1. Throw 1,000 numbers, output the histogram.

2. Throw 10,000 numbers, output the histogram.

3. Throw 100,000 numbers, output the histogram.

4. Throw 100,000 numbers with a 6x6 two-dimensional histogram (36 bins).

5. Throw 1,000,000 numbers, output the histogram.

6. Throw 10,000,000 numbers, output the histogram.

Use `qword` rather than `int` for variables to handle large counts exceeding 32,767. Remember, when creating histograms, you do not need to store all generated numbers—only keep track of the counts in each bin to optimize memory usage.

Ensure that bins are assigned so that every number in the range is equally likely to land in any bin — specifically, that 10 and 36 divide evenly into 360000, ensuring uniform distribution. Debugging output to the screen periodically can be helpful but should be minimized during large-scale simulations to improve performance. Consider the law of large numbers: as the number of trials increases, the observed distribution should approximate the theoretical uniform distribution. Explain any discrepancies observed.

Your program must generate the histogram data but not include the numbers themselves. Instead, your task is to output the histogram counts into text files. After all simulations, visualize the results in Excel or similar tools.

This assignment emphasizes efficient code structure: include proper comments, define constants, employ procedures and functions, and use `qword` variables for large counts. Internal comments should clarify logic, and the opening comment must specify the program's purpose, your name, date, and any file handling instructions.

Submitted code must compile successfully with the Free Pascal compiler, be original work, and follow coding best practices. The program must handle high numbers efficiently and produce reasonable, interpretable results related to the uniform distribution of random numbers and the law of large numbers.

Sample Paper For Above instruction

Empirical Random Number Distribution using Free Pascal

The purpose of this paper is to demonstrate how to explore the behavior of the Pascal random number generator through simulation and histogram analysis, as outlined in the CSC 540 assignment. The primary goal is to generate uniform random integers within a specified range, bin the results into histograms, and analyze their distribution to verify if they align with theoretical expectations based on the law of large numbers.

The study begins with the generation of three initial simulations: 1,000, 10,000, and 100,000 random numbers. These simulations help in understanding the behavior of the pseudo-random number generator across different sample sizes. The random numbers are uniformly distributed over the range 20001 to 380,000, with bins created to partition this interval into ten equal segments. Each segment serves as a histogram bin where counts are accumulated without storing individual numbers, optimizing memory and computational efficiency. The counts are then outputted to a text file for further visualization in Excel, where histograms are plotted to compare distribution patterns.

In addition to the one-dimensional histograms, a more complex two-dimensional histogram with 6 rows and 6 columns—totaling 36 bins—is constructed when simulating 100,000 random numbers. This grid provides insights into potential clustering or uneven distribution in a lower-resolution visual format. It effectively simulates a reduced-size chessboard grid, allowing for easier visual interpretation of distribution uniformity.

For larger sample sizes, the simulations extend to 1,000,000 and 10,000,000 numbers. These large datasets are particularly important because they more accurately reflect the expectation of uniform randomness, reducing statistical fluctuations. By observing the histograms from these larger samples, the experiment tests the law of large numbers: as the number of trials increases, the distribution of numbers per bin should approach a uniform distribution, with each bin containing roughly the same number of hits within statistical variation.

Throughout, the program emphasizes efficiency. Instead of storing all generated numbers, only the counts are kept in an array to prevent excessive memory use. To ensure uniform binning, the range was divided evenly such that each bin represents an equal portion of the total possible number of outcomes. Debugging print statements are included cautiously; during extensive runs, output is minimized to >speed up calculations. The code utilizes `qword` variables to handle large counts exceeding standard integer limits, ensuring the program can manage big data sets without overflow.

The output consists of text files with histogram counts, which can be imported into Excel. Visualization of the histograms confirms whether the random number generator behaves as expected. Ideally, the counts in each bin will be similar across large samples, illustrating the Law of Large Numbers in action. Any deviations are attributable to inherent statistical fluctuations. This simulation-based approach provides empirical validation for the randomness properties of the Pascal generator and illustrates fundamental concepts of probability and statistical convergence.

In conclusion, employing efficient programming techniques and careful statistical analysis, the simulation confirms that with sufficiently large sample sizes, uniform random number generators exhibit distribution characteristics predicted by theory. This exercise emphasizes the importance of large datasets in empirical randomness testing and practical applications involving stochastic processes.

References

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• Wikipedia contributors. (2023). Pseudorandom number generator. Wikipedia. https://en.wikipedia.org/wiki/Pseudorandom_number_generator
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