Monte Carlo simulations using Matlab

Explore Monte Carlo methods in MATLAB by learning what Monte Carlo simulations are, how to implement them with flow charts and pseudocode using random numbers to reveal statistical trends.

Explore the theory of Monte Carlo simulations and illustrate a dice-toss example to estimate the probability of rolling seven, including sequential versus non-sequential approaches.

Generate a discrete uniform random distribution by drawing integers from one to five, build a 100000-sample vector, and verify the distribution with a histogram, mean, and median.

Build a Matlab monte carlo script to simulate rolling two six-faced dice, estimate the probability of getting seven, and observe convergence toward the theoretical 0.1667 as iterations grow.

Learn how confidence intervals bound the Monte Carlo mean at 95 percent confidence level using the standard error, guiding sample size to improve convergence and simulation quality.

Learn to compute the mean and standard deviation of a success/failure distribution, and construct 95% confidence intervals to gauge convergence of Monte Carlo results in Matlab.

Visualize how Monte Carlo simulations converge in Matlab by plotting the confidence interval behavior across increasing sample sizes, while computing means, standard deviations, and 95% upper and lower bounds.

Explore the Monty Hall problem with three doors and a switch option; a car hides behind one door. Use the Monte Carlo method to decide whether to stay or switch.

Explore Monte Carlo simulations in Matlab to answer the Monty Hall stay strategy by simulating random door choices, counting wins, and estimating the average winning rate.

Students explore how a Monte Carlo MATLAB program estimates the Monty Hall stay probability, studying confidence interval convergence as sample sizes 100, 1000, and 100000 increase.

Demonstrate a Monte Carlo simulation in Matlab to answer the Monty Hall problem using the switch strategy, showing switching yields 2/3 winning probability.

Explore the Monty Hall problem through a Monte Carlo simulation in MATLAB, showing that switching doors yields about 66 percent winning probability while staying yields 33 percent.

Estimate pi with Monte Carlo simulations in MATLAB using a 2D uniform distribution, sampling points in a square with an inscribed circle, and derive a confidence interval.

Explore generating a continuous uniform distribution in MATLAB using rand to produce numbers between a and b in 1d, and compute its mean and variance.

Generate a two-dimensional uniform distribution with Matlab's rand, map to [a, b], and create 10,000 samples to illustrate Monte Carlo sampling for estimating area and pi.

Explore how Matlab Monte Carlo simulations verify a continuous uniform random distribution by measuring point density in subregions, linking density to area, and discussing sample size, memory, and accuracy.

In MATLAB, estimate pi with Monte Carlo by sampling random points in a unit square, counting inside a quarter circle, and computing pi ≈ 4*(inside/total); explore convergence and confidence intervals.

Apply confidence intervals to a Monte Carlo pi estimation in MATLAB by simulating points in a square and quarter circle, computing the mean, and deriving confidence bounds.

Explore Monte Carlo integration theory in Matlab by using uniformly distributed random samples to estimate the area under a curve, handling positive and negative regions with reference areas.

Select a control area for monte carlo simulations in matlab and estimate the area under the curve by counting uniformly distributed samples below it.

Generate samples within the control area and select those below the curve to estimate the area under f with Monte Carlo methods in Matlab, demonstrating convergence with more samples.

Explore Monte Carlo integration in Matlab to approximate area under curves when f(x) is negative, and match trapezoidal results by counting appropriate positive and negative points.

Apply confidence intervals to Monte Carlo integration in MATLAB by incrementally increasing sample sizes, computing the mean and standard error, and observing convergence of upper and lower bounds.

Explore reliability, availability, and maintainability through a water pump case study, using Monte Carlo simulations to compare purchasing an extra pump versus increasing maintenance resources to reduce downtime.

Introduce reliability, availability, and maintainability concepts and Monte Carlo with Matlab for decision making. Define reliability as uptime probability and maintainability as repair recovery.

Explore how reliability concepts link to failure times through the pdf and cdf, and use MATLAB to generate data following the exponential distribution via quantile function in Monte Carlo simulations.

Learn to model times to failure with an exponential distribution in Matlab, fit parameters from historical data, and use Monte Carlo simulations to estimate reliability, availability, and maintainability.

Monte Carlo simulations using Matlab to simulate asset times to repair with an exponential distribution, estimate mean time to repair and time to failure, and support a reliability study.

Develop a Matlab script that builds a binary status vector of up and down times from times to fail and repair, and plots two cycles to assess system availability.

Monte Carlo simulations in Matlab compare adding a second pump versus reducing mean time to repair to maximize system availability, showing a rise from 87% to about 99%.

Explore RAM analysis for power systems using Monte Carlo and Matlab, expanding on prior Monte Carlo methods to reliability, availability, and maintainability and reliability centered maintenance.

Celebrate completing Monte Carlo simulations using MATLAB by applying the course's foundational principles to diverse problems and exploring future possibilities.