R Programming for Simulation and Monte Carlo Methods

Practice crafting R expressions with x, a, and b; study operator precedence, exponentiation, and assignment; work with vectors, matrices, sequences, and identity matrices.

Explore R expressions through practice with vectors, modulo operations, and logical indexing to exclude values divisible by 2, 3, or 7, and manage queues with length and which for subsetting.

Explore Monte Carlo simulation with a coin-toss game to estimate Peter's cumulative winnings over 50 tosses using sample, replicate, and cume sum.

Use a simulation approach to run four 50-toss games, plot cumulative winnings, and compare exact binomial probability of breaking even with a reproducible set.seed run.

Learn to implement Monte Carlo experiments in R by writing a named function for one game, then repeat simulations to estimate fortune, lead frequency, and break-even probabilities.

write a simulation function in r by using the last expression as return, replicate trials, tabulate results with table, plot frequencies, and compute breaking-even probability via binomial.

Explore a Monte Carlo coin-toss simulation to approximate exact probabilities, track the final fortune, count lead changes, and identify the maximum cumulative winnings.

Run a thousand coin-toss simulations in R, storing f, l, and m to analyze lead frequency, and plot the distribution of maximum winnings in 50 plays.

Simulate random hat permutations to count correct matches using boolean vectors and 0/1 sums; explore the distribution of matches and the mean as the number of hats grows.

Learn to use R to simulate the hat permutation problem: create an ordered hat vector, generate a random permutation with sample, and count matches with the original order.

Investigate the hat problem through random permutations, converting booleans to ones and zeros, counting matches with a user defined R function, and examining how hat count affects matches.

Explore the hat problem through Monte Carlo style simulations by replicating random permutations of 10 hats, tallying matches, and estimating the distribution and expected value (mean) of matches.

This lecture demonstrates simulating random hat permutations to estimate the probability of no matches across 2 to 20 hats; uses apply and replication to compute and plot the expected value.

Use R to simulate the hat-check problem by representing hats with 1s and 2s, running 1000 trials to estimate matches, probability of 10+ matches, and expected matches.

Perform a Monte Carlo simulation in R of 20 hats in two colors, using scramble hats and replicate function to estimate matches and probability of at least 10 correct colors.

Simulate collecting a complete set of baseball cards using R's sample function to draw 20 cards from 10 players, with and without replacement, and count unique players.

Simulate a complete set of 586 baseball cards by sampling with replacement. Implement a collector function in R to sample 3000 cards and estimate the probability using Monte Carlo.

Use a Monte Carlo simulation to find the optimal number of baseball card purchases, balancing random five-cent packs and twenty-five-cent dealer cards to minimize total expected cost.

simulate buying baseball cards in R to estimate total cost using 500 trials and replicate to compute the expected cost across purchase levels, with a minimum near 950 cards.

Attempt the collecting state quarters exercise and recognize you may need concepts from later baseball cards videos; a solution will be posted later in the course.

simulate buying 100 quarters from 50 states with replacement, measure unique states, replicate results, and estimate 43 1/3 unique coins and about $13.26 to complete the set.

Explore streaky batting behavior with a Monte Carlo approach; assess the significance of hitting streaks and build a function to compute run lengths in binary sequences using a padded vector.

Compute zero-base-hit sequences in game data, derive streak lengths from consecutive zeros, filter to nonzeros, and determine the longest streak with a custom function.

Explore a nonparametric bootstrap approach to baseball batting streaks by permuting 116 at-bats, estimating the distribution of the longest streak, and assessing significance with Monte Carlo simulations.

Explore counting switches in binary sequences, compute the number of switches with a difference-based method, and compare observed switches to a 10000-run Monte Carlo simulation in R.

Assess Utley's 35-game hitting streak with a non-parametric permutation of 116 hits in 160 games, and analyze switches via diff and histogram across thousands of samples to judge unusual behavior.

Use Monte Carlo simulations with independent uniform arrival times for Sam and Amy to estimate the probability Amy arrives before Sam, visualized via a scatterplot and area region.

Compute the mean of the differences between each pair, defined as aney's arrival minus sam's. The mean is 0.479 hours, with a small standard error; 25% arrive earlier.

Apply trimmed means to reduce outlier effects in Monte Carlo inference, compute the mean squared error and its standard error from simulated samples.

Apply Monte Carlo inference with median-based summaries for non-normal data, compute standard errors and confidence intervals, and illustrate an upper confidence limit for variance from normal samples.

Extend Monte Carlo simulations to estimate upper confidence limits for sigma squared and assess empirical coverage under normal and chi-squared populations, including bootstrapping and jackknife ideas.

Apply Monte Carlo simulation to estimate the population median theta of an exponential distribution using 1000 samples of size 21 and compare the median's distribution to the exact density.

Use Monte Carlo simulation to compare two estimators for the total number of taxis: the maximum of observed cabs and twice the sample mean, under a uniform population.

Compare two estimators for the taxi count, max of five observations versus twice the mean, using Monte Carlo simulations to assess bias, standard error, and absolute error via box plots.

Use a Monte Carlo simulation to estimate expected total travel time over a five-day week, probability of at least one late class, and longest travel time with its standard error.

Use Monte Carlo simulation to estimate a student’s travel time and lateness probability, with normal times mean 20 and standard deviation 4 over five days, and find longest travel time.

Map from the uniform distribution to non-uniform distributions using f(x), with histograms, scatter plots, and autocorrelation to validate unbiased Monte Carlo sampling.

We demonstrate that applying the inverse transform to a uniform sample generates exponential random variables, with x equal to the negative logarithm of u. Using 10,000 simulations, we compare histograms of the uniform-derived and directly generated exponential samples to show similar shapes.

Simulate discrete random variables by using a uniform input to generate binomial outcomes, and derive the cumulative distribution function and probability mass function to model discrete data in R.

simulate a binomial variable with a homemade generator and recursive function, using 100 trials at 50% success, and compare the Monte Carlo histogram to the binomial function for confidence intervals.

Plot the actual and simulated binomial distributions from 10-event trials with 70 percent probability, then validate with a 95% confidence interval showing close agreement with real values.

Apply the Newton-Raphson method to find roots by iterating with the first derivative, starting from an initial guess and a tolerance, guarding against divergence with a maximum iterations cap.

Generate random variables using a binomial-based approach in R, plot a histogram, compare it to the binomial mass function, and validate the transform against the actual variable.

Learn to simulate a binomial random variable in R with n=25 and p=0.2, generate 5000 samples, plot a histogram, and compare with the theoretical probability mass function.

Explore how to simulate random variables with uniform sampling, compare density and cumulative distribution estimates to theoretical values, and evaluate performance between loop-based and direct transformation methods in R.

Learn to simulate exponential and chi squared distributions from a uniform distribution using inverse transforms, compare methods, and assess performance.

Investigate transforming known functions to simulate samples, note the difficulty of obtaining the standard normal from uniform input, and apply the inverse method to discrete distributions.

Simulate Poisson distributions with large lambda using a 3*sqrt(lambda) window and 10,000 trials. Compare a uniform-based simulation to the native R function.

Explore the accept-reject method for continuous distributions by simulating from a uniform rectangle and accepting points under the target density, using the beta distribution with shape parameters a and b.

Apply the acceptance–rejection method to simulate the area under a triangle by partitioning the interval into ranges with f(x)=x on [0,1] and f(x)=2−x on [1,2], and plot a histogram.

Explore the Poisson distribution as a discrete random variable, its lambda-driven probability and cumulative density functions, and how to simulate Poisson outcomes from uniform inputs.

Explore simulating a Poisson distribution by transforming uniform and exponential random variables, compare performance between uniform-based, exponential-based, and vectorized native functions, and assess efficiency over thousands of trials.

Use the trapezoidal rule to estimate the area under f(x)=4x^3 on [0,1]. Create a function accepting f, a, b, and n to compute width and sum trapezoids.

Use Simpson's rule to approximate the area under f(x) from a to b with parabolic segments, exact for cubic functions and more accurate than trapezoidal for polynomials.

Apply simulation techniques to estimate the area under a curve between a and b using trapezoidal and Simpson's rules. Explore positive one-dimensional cases and real case studies like disease spread.

Learn to estimate definite integrals in R by using the trapezoidal rule with subintervals, computing x and f(x) values, and summing the areas, then apply Simpson's rule for quadratic accuracy.

Explore Simpson's rule with parabolic interpolation to approximate the area under f from a to b, using midpoint subintervals on even and odd segments and h/3 scaling.

Compute the area under the normal curve using little phi and big Phi, applying Simpson's rule to estimate the cumulative distribution across z values.

Derive z-values from tail areas of the phi function by reversing the computation of the pdf and cumulative area, using provided functions to determine significance and confidence levels.

demonstrates hit-and-miss monte carlo integration to estimate the area under a curve, using uniform sampling over a bounding rectangle and extending to two dimensions with negative values.

Generate uniform x and y samples and use a hit-or-miss test against f to estimate the integral. The approach works but needs many simulations, showing limited efficiency.

Explore in R how to invert the normal cumulative distribution function using phi functions and Newton-Raphson, converting area under the curve into z scores and standard deviations.

Explore permutation tests and notched box plots to detect median differences, noting when medians fall outside notches and small-sample limits; verify with a six-way anova on weight, and residual checks.

Perform a permutation test to compare two unbalanced groups by pooling observations, generating random splits, and evaluating mean differences to assess significance beyond a t-test.

Learn how to perform permutation tests to compare distributions in R, using the Chckwts example to assess mean differences with KS and exact tests, plus simulation-based p-values.

Develop skills in permutation tests, K-S test distributional summaries, compare groups, and implement an exercise to analyze correlations with normal and lognormal data in R.

Explore implementing an exact and permutation-based Spearman rank correlation test in R, comparing exact p-values to simulation-based p-values across normal and lognormal data.

Build two object oriented classes in R and run seed dispersal simulations to demonstrate polymorphism, generic functions, and mean or variance methods across S3 and S4.

Explore seed dispersal analysis by building a trap transect class in r, with a constructor function and methods to manage distances, trap area, and seed counts in a transect.

Create a trap transect class in R, implement a constructor, and define print and mean methods to summarize weighted seed dispersal along a transect.

Simulate the radial distribution of seed rain around a plant from transect data using rejection sampling and gamma or exponential decay models.

Compare gamma density (rate 1/2) with an exponential density for seed dispersal around a plant, then use rejection sampling with exponential draws and a uniform threshold to estimate radial density.

Compare lognormal and weibull transect density models for seed dispersal, and validate simulated radial dispersal with acceptance–rejection sampling against exact radial PDFs; build functions to simulate transects and radial dispersal.

Create a transect holder object to model the seed shadow around a plant, fit transect distances to a nominated distribution (exponential, lognormal, gamma, Weibull) using the SDR function in Mass.

Explore building and testing a generic simulate function for transect holder objects and perform Monte Carlo seed dispersal simulations using Weibull distributions, q-q plots, and acceptance rejection sampling.