Virtual Laboratories in Probability and Statistics

The goal of the Virtual Laboratories in Probability and Statistics is to provide free, high quality, interactive, web-based resources for students and teachers of probability and statistics. Basically, our project consists of an integrated set of components that includes expository text, applets, data sets, biographical sketches, and an object library.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the ballot experiment: The votes in an election are randomly counted. The event of interest is that the winning candidate is always ahead in the vote count.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of Bertrand's experiment: a random chord on a circle The event of interest is whether the length of the chord is larger than the length of the inscribed equilateral triangle. Three models for generating the random chord can be used.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet illustrates Bayesian estimation of the probability of heads for a coin. The prior beta distribution, true probability of heads, and the sample size can be specified. The applet shows the posterior beta distribution.

This resource consist of a Java applet and expository text. The applet ...

This resource consist of a Java applet and expository text. The applet simulates Bernoulli trials in terms of coin tosses. The random variables of interest are the number of heads and the proportion of heads. The number of coins and the probability of heads can be varied. The applet illustrates the law of large numbers and the central limit theorem.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates Bernoulli trials in terms of random points on a timeline. The random variables of interest are the number of successes and the proportion of successes. The number of trials and the probability of success can be varied. This applet illustrates the law of large numbers, the central limit theorem, and the binomial distribution.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the birthday experiment: a sample of size n is chose at random and with replacement from the first m positive integers. The random variable of interest is the number of distinct sample values. The event of interest is that all sample values are distinct.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates the bivariate normal distribution. The means are set at 0, but the standard deviations and the correlation can be varied. Simulated points from the distribution are shown as dots in a scatterplot.

This resource consists of a Java applet and expository text. The Java ...

This resource consists of a Java applet and expository text. The Java applet illustrates the bivariate uniform distribution on three types of regions: a square, a circle, and a triangle. Simulated points from the distribution are shown as dots in a scatterplot.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates Buffon's coin experiment. The radius of the coin can be varied. The applet illustrates a random experiment, the sample space, random variables, events, probability, and relative frequency.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates Buffon's needle experiment and the corresponding approximation of pi. The event of interest is that the needle crosses a crack. The length of the needle can be varied. The applet illustrates a random experiment, the sample space, random variables, probability, and relative frequency.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of drawing n cards from a standard deck. The parameter n can be varied.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates the chuck-a-luck experiment of rolling 3 fair dice. The random variable of interest is the net profit of the player.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the experiment that consists of tossing a coin and then rolling either a red die or a green die, depending on the outcome of the coin toss. The probability of heads and the distributions of the two dice can be specified. The applet illustrates a two-stage experiment.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates coin tosses. The number of coins and the probability of heads can be varied.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the coupon collector problem, which consists of sampling from the first m positive integers, with replacement, until k distinct values are obtained. The parameters m and k can be varied. The random variable of interest is the size of the sample.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the standard casino craps game. The random variables of interest are the winning under various bets.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates rolling n dice. The random variables of interest are the sum of the scores, the maximum score, the minimum score, and the number of aces. The number of dice and the probability distribution that governs the dice can be specified. The applet illustrates various distributions and the central limit theorem.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates the experiment of rolling a die and then tossing a coin the number of times shown on the die. The die distribution and the probability of heads can be specified. The applet illustrates a two-stage experiment.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the experiment of selecting n objects at random from the first m positive integers. The random variables of interest are the order statistics. The applet illustrates the distributions of the order statistics.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the fire process, an interacting particle system consisting of a rectangular lattice of "trees". At each discrete time unit, each tree is either on green, on fire, or burnt. A tree on fire at a given time can catch neighboring trees on fire at the next time unit. The probabilities of fire spread can be varied.

This resource consists of a Java applet and descriptive text. The applet ...

This resource consists of a Java applet and descriptive text. The applet illustrates Bernoulli trials in terms of a Galton Board. The number of trials and the probability of success can be varied. The random variables of interest are the number of successes and the probability of a success. The applet illustrates the binomial distribution, the law of large numbers and the central limit theorem.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet illustrates the Galton board in which a ball falls through a triangular array of pegs. Various combinatorial ideas are illustrated, including combinations, bit strings, and binomial coefficients.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates a random sample from a gamma distribution, and computes standard point estimates of the shape and scale parameters. The bias and mean square error are also computed.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates the time of the k'th arrival in a Poisson process. The arrival number k and the rate of the Poisson process can be varied. The applet illustrates the gamma distribution and a special case of the central limit theorem.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet allows the user to construct a histogram by clicking on a number line to generate the sample data. The class width can be varied and various summary statistics can be displayed.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet allows the user to construct a histogram by clicking on a number line to generate the sample data. The graph of an error function is shown, either mean square error or mean absolute error.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet allows the user to construct a scatterplot by clicking on a graph to generate the bivariate data. The means, standard deviations, correlation, and the regression line are shown.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the matching experiment, which consists of a random permutation of the first n positive integers. The random variable of interest is the number of matches (the number of integers in the random permutation that are in their correct positions). The applet illustrates the distribution of the number of matches and the convergence to the Poisson distribution.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet is a simulation of the Monty Hall experiment: a car is behind one door, goats are behind the other two doors. The player chooses a door and then the host opens another door. The player is given the option of switching to the remaining door. The stochastic behavior of the host and the probability of the player switching can be specified.

This resource consists of a Java applet and descriptive text. The applet ...

This resource consists of a Java applet and descriptive text. The applet illustrates the number of trials needed to get a specified number of successes in Bernoulli trials, in terms of random points on a discrete timeline. The applet illustrates the negative binomial distribution. The number of successes and the probability of success can be varied.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates a random sample from a normal distribution, and computes standard point estimates of the distribution mean and standard deviation. The bias and mean square error are also computed.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates the order statistics of a random sample from a given distribution. The sample size, order, and sampling distribution can be specified.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates a random sample from a Pareto distribution, and computes standard point estimates of the shape parameter. The bias and mean square error are also computed.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates the arrivals in a Poisson process during a time interval. The length of the interval and the rate of the process can be varied. The applet illustrates the Poisson distribution and a special case of the central limit theorem.

This resource consists of a Java applet and expository text. The experiment ...

This resource consists of a Java applet and expository text. The experiment consists of rolling 5 fair dice. The random variable of interest is the type of hand.

This resource consists of a Java applet and Expository text. The applet ...

This resource consists of a Java applet and Expository text. The applet simulates the probability plot that compares the empirical quantiles of a sample from a sampling distribution to the distribution quantiles of a test distribution. The sampling distribution, test distribution, and sample size can be specified.

This resource consists of a Java applet and descriptive text. The applet ...

This resource consists of a Java applet and descriptive text. The applet illustrates the problem of points: a coin is tossed until either n heads occurs or m tails occurs. The parameters n, m, and the probability of heads can be varied.

This resource consists of a Java applet and descriptive text. The applet ...

This resource consists of a Java applet and descriptive text. The applet illustrates a random walk on a discrete time interval from 0 to n. The time n and the probability of a step to the right can be varied. The random variables of interest are the final position, the maximum position, and the time of the last zero.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates the red and black experiment in which a player bets on a sequence of Bernoulli trials until a target fortune is reached or the player is ruined. The initial fortune, target fortune, and trial win probability can be varied, and the user can select either of two basic strategies: timid play or bold play.

This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet allows the student to play the game of red and black, by betting on a sequence of Bernoulli trials until either a target is reached or the player is ruined. The target, initial fortune, and trial win probability can be varied.

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