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Units, Constants and Useful Formulas

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Last modified: January 26, 2018

Monte-Carlo Methods ------------------- The Monte Carlo method is just one of many methods for analyzing uncertainty propagation, where the goal is to determine how random variation, lack of knowledge, or error affects the sensitivity, performance, or reliability of the system that is being modeled. Monte Carlo simulation is categorized as a sampling method because the inputs are randomly generated from probability distributions to simulate the process of sampling from an actual population. So, we try to choose a distribution for the inputs that most closely matches data we already have, or best represents our current state of knowledge. The data generated from the simulation can be represented as probability distributions (or histograms) or converted to error bars, reliability predictions, tolerance zones, and confidence intervals.   ------   | | ---x1--| |---y1 ---x2--| f(x) | ---x3--| |---y2   | |   ------ 1. Create a parametric model y = f(x1,x2,...xn) 2. Generate a random set of inputs x1,x1,...xn 3. Evaluate the model and store the results as yn 4. Repeat steps 2 and 3 for i = 1 to n 5. Analyze the results using histograms, summary statistics, confidence intervals, etc. Example: Compute the value of π Inscribe circle inside square with sides x = 1, y = 1. Consider upper right quadrant: y | |. | . 1 | . | . | . ------------ x 1 Area of quadrant = π/4 Area of square = 4 Generate random combination of x and y between 0 and 1. From Pythagoras: if √(x2 + y2) <= 1 then count as a 'hit'. if √(x2 + y2) > 1 then count as a 'miss'. 4 * hits π = --------------- (hits + misses) It is simple to write a simple javascript program to do this: var hits = 0; var i = 0; var pi = 0; var x = 0; var y = 0; var hyp = 0; for(i=0; i<=1000000; i++) { x = Math.random(); y = Math.random(); hyp = Math.sqrt((x*x)+(y*y)); if(hyp <= 1) { hits++; } } pi = (4*hits)/i; compute π