> # SUMMARY OF CURRENT PROBABILITY AND STATISTICS MODULES

> #### Biological Phenomena Described Using Statistics and Probability (Jonathan P. Zehr)

> The Biology Links module provides an interactive discussion of biological, ecological and evolutionary phenomena that are based on statistics and probability. The module is being developed as a web page which will have several links to subtopics that will be used in different biology classes. The initial page can be seen at http://www.rpi.edu/~zehrj/links/links.html. This preliminary page focuses on evolution, and how the probability of mutation events can be used to compare DNA and protein sequences to determine the evolutionary relationship of organisms. An interactive module is being developed that allows the student to simulate evolution and change the parameters of evolution and observe the effects on evolutionary relationship of molecular sequences.

> #### 1D Random Variables (Elisa Barney-Smythe)

> A visual introduction to 1D random variables. User has ability to select between Uniform, Gaussian/Normal and Exponential Probability distributions. Also the parameters for these distributions [(a,b), (m,s), (alpha)] are selectable. A plot showing the sample points from a distribution as defined are plotted. A frequency histogram is drawn based on the samples, and the theoretical pdf is drawn on the same axis. The cummulative probability distribution is also drawn, both as a histogram and as the theoretical plot.

> #### 2D (Independent) Random Variables (Elisa Barney-Smythe)

> A visual introduction to 2-D random variables. User has ability to select between Uniform, Gaussian/Normal and Exponential Probability distributions for both the X and the Y parameter. The parameters for these distributions [(a,b), (m,s), (alpha)] are selectable. A plot showing the sample points from a distribution as defined are plotted as x's with the x-position governed by the value of the X random sample and the y-position governed by the value of the Y random sample. A histogram of the marginal probabilities is drawn based on the samples, and the theoretical marginal distribution is drawn on the same axis. The marginal cummulative probability distribution is also drawn, both as a histogram and as the theoretical plot. A 2-D mesh showing either the theoretical 2D pdf or a sample 2D histogram of the 2D pdf is shown with user controls to rotate and tilt the surface.

> #### Gaussian Error Functions (Elisa Barney-Smythe)

> Module is about converting a N(m,s) distribution to a N(0,1) distribution to use Gaussian CDF tables. The user selects the mean (m) and the standard deviation (s) of a gaussian. Then the user can choose between Phi(x), 1-Phi(x) or Phi(x)-Phi(y) with user selectable x & y. A plot of the probability (area) indicated by the choice of x & y on a N(m,s) curve will be drawn. The corresponding probability will be shown too.

> #### Geiger (Elisa Barney-Smythe)

> Module shows 3 scenarios: Geiger counter bleeps, cars passing a toll booth and phones ringing at a phone exchange. Samples are generated with a poisson distribution. User has option for looking at time between events (exponential distribution) and number of events in an interval (poisson distribution) User must mark the events in the window showing sound bleeps (or cars or phones) until 10 have been marked then auto mark is allowed. A histogram of time between marked events is shown, with sample mean and corresponding theoretical exponential dist. Under number of events in interval, interval length can be set. Sample distribution for this is shown also (both pmf & CDF).

> #### Poisson (Poisson/Exponential Relationships) (Elisa Barney-Smythe)

> Allows generation of N samples which are from a poisson/exponential process. Three parameters: lambda (parameter for exponential), T (length of time interval in Poisson) and thus alpha=lambda*T (parameter for Poisson) are user selectable. 4 different displays analyzing the data exist: 1) Plots of pdf & CDF of time between events (exponential), 2) "Exponentialness Plot" Plot of log complementary cumulative distribution vs x of time between events, 3) Plots of number of events in intervals of user selectable length (poisson), and CDF for poisson, 4) "Poissonness Plot" plot of ln(Pk(k)*k!) vs k (number of events in intervals). For plot 1, a histogram showing time between events is shown as well as the cumulative occurrences of time differences less than given time and corresponding theoretical exponential distribution, also sample mean and standard dev. For plot 3, a stem plot showing count of number of intervals with k events and another plot with number of intervals with k or fewer events. The E[k] is shown and the theoretical distribution for this value are plotted on same plot. Window showing event locations graphically is shown with data viewing controls.

> #### Joint Gaussian Distributions (Elisa Barney-Smythe)

> A visual introduction to 2-D Gaussian random variables. X & Y are joint gaussian random variables. The parameters for the means (mx & my), standard deviations (sx & sy) and the correlation (p) are selectable. The number of sample points is user defined. (may be zero) A plot showing the sample points from a distribution as defined are plotted as x's with the x-position governed by the value of the X random sample and the y-position governed by the value of the Y random sample. The equiprobabuilty contours are drawn on the same axis. The 2D probability density function surface is drawn based on the parameters selected. The view (angle & elevation) of this surface can be selected by the user. Eventually (but not coded and available for suggestions & modification) the conditional probabilities as x is varied (or y) will be shown as a movie. The marginal of that variable will be shown also. This module is currently under construction. Completion is expected by the end of January 1997.

> #### Conditional Prob Networks (Elisa Barney-Smythe)

> A visual introduction to conditional probability via a communication channel where the input symbol will be changed to another symbol with a certain probability. The number of input symbols will be user selectable. The number of output symbols will be user selectable. The conditional probabilities will be user selectable The initial symbol will be shown and the histogram of input probabilities will accumulate. The path between input and output symbol will be graphically traversed and the final symbol probabilities will accumulate. This module is currently under construction in Java by a URP. Completion is expected by May 1997.

> #### Great Expectations (Elisa Barney-Smythe)

> A visual introduction to Expected value. A distribution (both discrete & continuous) will be shown. The user guesses the E{x} and Var{x}. The values are shown graphically, and the correctness factor of the guess will also be displayed. The complexity of the distribution will vary from simple to complex, discrete, piecewise linear, and common non-linear distributions. This module is currently under construction. It is still only in its conceptual stages, and its completion date is unknown.

> #### Probability and Computer Networks (Ken Vastola)

> Our focus is on using the popular topic of computer networks to motivate students to learn probability using the World Wide Web. Java applets have been developed to animate various network components. The impact of randomness and the need to model it using probability is stressed. Also teaches important properties of Poisson and related distributions visually as well as mathematically.

> #### Yield modeling of IC Chips (M. Raghavachari)

> This module discusses an application of Poisson distribution and exponential distribution to yield modeling of IC chips. It illustrates how the final negative binomial yield model emerges by a combination of Poisson distribution for the number of defects and exponential distribution for the defect density. It also brings out the concept of mixing of two distributions.

> #### Poisson Distribution Basics (M. Raghavachari)

> This module explains how the number of defects on a chip tends to follow a Poisson distribution as a limit of Binomial distribution.

> #### Applications of the Bernoulli and Binomial Random Variables (Wayne Roberge)

> This module will develop a suite of exercises that require the students to apply the Bernoulli and binomial variables to simple physical processes that can be modeled as random walks. The module is under development; a tentative list of the exercises is: 1. Quantum Mechanical Tunneling on a One-Dimensional Lattice 2. Diffusion with Gravitational Settling 3. Evaporation of a Star Cluster 4. Diffusion-Limited Reactions on Surfaces 5. Poisoning of Catalysts

> #### What Concepts in Probability Can We learn From Internet Traffic? (Chuanyi Ji)

> Internet traffic has been found to possess very different statistical properties from those due to Poisson processes. Such properties are the so-called self-similar properties. One of such properties can be characterized by the long-range dependence in the auto-correlation function of the traffic. The auto-correlation function decreases at a slow (polynomial) rate with respect to lags. The auto-correlation due to Poisson processes, however, decreases at a much faster (exponential) rate in terms of lags. These facts provide interesting examples for students to learn about Poisson processes and auto-correlation functions through processing Internet data.

> The structure of this module has been planned as follows. (a) A link will be provided to a Poisson module developed already by a group member. Students can first run this module to get to know the basics on Poisson processes. (b) Auto-correlation functions will be defined and evaluated from a Poisson process. (c) Auto-correlation functions will be evaluated from Internet traffic data, and compared with that from a Poisson process. (d) A simple description on the self-similar process will be given to explain the difference between the auto-correlation functions of the real traffic and the Poisson process, and why Poisson is not a good model for Internet traffic.