Studying the Binomial Distribution Using LabVIEW - Journal of

Laboratory Experiment pubs.acs.org/jchemeduc

Studying the Binomial Distribution Using LabVIEW Danielle J. George and Nathan I. Hammer* Chemistry and Biochemistry, University of Mississippi, University, Mississippi 38677, United States S Supporting Information *

ABSTRACT: This undergraduate physical chemistry laboratory exercise introduces students to the study of probability distributions both experimentally and using computer simulations. Students perform the classic coin toss experiment individually and then pool all of their data together to study the effect of experimental sample size on the binomial distribution. Simulations of the coin toss experiment are also performed using the software package LabVIEW. LabVIEW facilitates the creation of complex computer programs in a short period of time, even by beginner programmers. Histograms in LabVIEW are displayed in real time with students adjusting the number of simulated coin tosses on the fly using a virtual knob or a slider, up to 1 million individual trials. This allows the students to see firsthand the evolution of the binomial distribution into the Gaussian distribution with a large sample size. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Laboratory Instruction, Hands-On Learning/Manipulatives, Laboratory Computing/Interfacing, Statistical Mechanics

■

another, and the probability of a “success” (in this case heads) is the same for each attempt or trial:

INTRODUCTION The undergraduate physical chemistry course is usually organized into four subgroups: thermodynamics, quantum mechanics, spectroscopy, and kinetics.1−4 Statistical mechanics permeates each of these topics as it relates the microscopic world to the macroscopic,5 and many chemists would likely agree that statistical thermodynamics is a fundamental topic in all subfields of chemistry. When learning about statistical mechanics, it is important for undergraduate students to understand simple distributions such as the binomial and Gaussian distributions. Here, we describe a physical chemistry laboratory exercise that combines both experimentation and simulation of the binomial and Gaussian distributions. Students perform the classic coin toss experiment6 where they randomly dispense 50 coins onto a surface and count the number of heads for a total of 50 individual Bernoulli trials. After repeating this exercise for a total of 10 experiments, they then simulate the process using the programming software package National Instruments LabVIEW. By viewing the resulting histograms using LabVIEW in real time, students can easily perform up to a million simulated experiments and see firsthand how the Gaussian distribution emerges from the binomial. The inclusion of LabVIEW introduces general-purpose programming to students at the undergraduate level and promotes creative problem solving that is necessary for professional or graduate level training. The binomial distribution is the exact distribution of a collection of Bernoulli trials.7,8 These Bernoulli trials have only two possible outcomes, such as the occurrence of either heads or tails when flipping a coin. Each trial is independent from one © XXXX American Chemical Society and Division of Chemical Education, Inc.

P(n) = pn (1 − p)N − n

N! , n! (N − n)!

n = 0, 1, 2, ..., N (1)

where N is the number of trials, p is the probability of each trial, and n is the number of successes. In the case of flipping a coin p = 0.5. In nature there are many processes that have two possible outcomes or situations where two options are possible. A good example is the case of chlorine isotopes having a mass of either 35 amu (p = 0.76) or 37 amu (p = 0.24). The Gaussian distribution is derived from the binomial distribution for a large number of trials.9 The familiar bellshaped curve10 that results is used for many different representations of data from business to science, but this distribution also reveals the random error in experimental measurements. These errors are the summation of many small, independent errors that occur from measurements in both directions of the vertex or the mean of the distribution. The Gaussian distribution is given by the equation p(x ) =

2 2 1 e−(x − μ) /2σ , σ 2π

−∞ ≤ x ≤ ∞

(2)

where σ is the standard deviation, μ is the mean, and p(x) is the probability of such event. There are many examples in the literature of computer software packages being used to assist in teaching physical

A

dx.doi.org/10.1021/ed500684k | J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

Laboratory Experiment

chemistry concepts.11−25 For example, the Fortran programming language was used four decades ago to simulate chemical phenomena using statistical mechanics.22 Coin toss experiments have previously been simulated using computer software packages to demonstrate entropy, introduce microstates, and demonstrate how systems will move toward equilibrium.25 National Instrument’s LabVIEW has been shown to be an excellent choice for controlling instrumentation and acquiring data,26−43 and for the analysis of experimental data.44−49 The use of programming software packages such as LabVIEW offers many opportunities in the classroom. Simulations in LabVIEW have been used to teach chemical concepts,50−53 to teach instrumental design,43,54,55 and to simulate data,56−58 without requiring an extensive set of physical resources. For example, a time-of-flight mass spectrometer has been simulated in LabVIEW,59,60 as well as a refractometer to understand the liquid−vapor equilibrium in a binary system.37 LabVIEW has also been used to simulate measurements of strong electrolytes.61 Students are known to learn in different ways. This laboratory exercise is designed to reach a set of diverse learners including spatial, kinesthetic, logical, intrapersonal, and interpersonal.62 Spatial learners are ones that benefit the most due to the exhaustive set of illustrations that communicate these concepts. Through illustrations, spatial learners are able to see the graphical data presented by the two distributions. Since students are tossing the coins and counting them, kinesthetic learners have the opportunity to physically be involved in the experiment. These students learn by doing and likely feel connected to the experimental component of this laboratory exercise. Logical learners are active within the procedure as well. These will benefit from the mathematical equations involving the binomial and Gaussian distributions. This laboratory exercise also benefits students that are intrapersonal and interpersonal learners. Interpersonal learners gain from the collaboration that is done in this laboratory from sharing one’s data with the class and seeing how one’s contributions add to the whole. Interpersonal learners also will gain from the solitary time spent counting their own number of heads developing their own LabVIEW simulation. This laboratory exercise has been performed by the physical chemistry laboratory class at the University of Mississippi for the past seven years (2008−2014) and is typically one of the students’ favorites. The student handout (Supporting Information) includes an introduction to the different features of LabVIEW so that students are prepared before the meeting of the laboratory. The exercise can be performed in time under 3 h and includes a prelaboratory lecture of approximately 40 min, the experimental coin toss exercise that takes about 20 min per student, with the remaining time devoted to exploring the creations of distributions using LabVIEW. Students can take turns generating their experimental data sets while working on their LabVIEW programs.

container onto a surface. Each head is counted as a success, and the total number of heads is recorded in their laboratory notebook. Each student individually performs this nine more times for a total of 10 trials. Students share their data with the entire class to create a larger sample size. After class, students calculate the mean and standard deviation of their set and the aggregated class set, plot the class distribution using a spreadsheet program such as Microsoft Excel, and compare their distribution to a Gaussian distribution using their mean and standard deviation as shown in Figure 1.

Figure 1. Spring 2014 physical chemistry laboratory course coin toss data (80 individual trials) compared to a Gaussian distribution calculated using the mean and standard deviation.

LabVIEW facilitates the creation of complex computer programs in a short period of time even by beginner programmers.43 In this laboratory exercise students start with a blank virtual instrument and start by first inserting the random number generator to produce a number between 0 and 1. By wiring the random number to round to the nearest integer (0 or 1) and to a numeric indicator, a Bernoulli trial is created with every run of the program and students can simulate individual coin flips as shown in Figure 2. The student then needs to toss 50 coins, and the easiest way to accomplish this task is by using a for loop. This process represents one experimental determination of the number of successes (heads). The number of heads is determined by summing the array that contains these individual trials as shown in Figure 3. Each time this program is run the experiment is simulated and the number of heads is displayed. In order to repeat the simulation a certain number of times, such as in the case of the actual student experiment, another for loop needs to be used, as shown in Figure 4. This time a variable control can be used for the number of experiments such as a slider bar that can be adjusted in real time. It is suggested that students compare the resulting histograms for 10, 100, 1000, 100,000, and one million trials. One million experiments can be performed in about 1 s on a modern PC. In order to visualize the results a histogram can be created which quickly summarizes the number of times each the number of heads is observed with the range of possibilities going from zero to 50. LabVIEW has a General Histogram icon that easily creates a histogram from the resulting array of experiments. Since 51 possible outcomes are possible in the experiment (zero heads up through 50 heads), the number 51 (or 50 + 1) is wired into the “number of bins” input and the numbers −0.5 and 50.5 into the minimum and maximum inputs so that bin centers of 0, 1, 2, ..., 48, 49, 50 will be calculated. The program

■

EXPERIMENTAL PROCEDURE AND EXAMPLE RESULTS At the University of Mississippi, the prelaboratory lecture for this exercise has two components: (1) a brief introduction to probability distributions and the importance of statistical mechanics in chemistry and (2) an introduction to using LabVIEW that includes the use of loops, the random number generator, arrays, and plotting histograms. Students start the laboratory exercise by individually dispersing 50 coins from a B




Figure 2. Simple LabVIEW program that generates a Bernoulli trial.

Figure 3. LabVIEW program that indicates the number of successes (heads) when dispersing 50 coins.

Figure 4. LabVIEW program that creates a histogram of successes (heads) with a slider bar that allows for the easy control of the number of simulated experiments.

can be run with one set of experiments at a time and capture screenshots of the distribution as a function of the number of experiments. By putting the whole program into a while loop with a STOP button the student can continually adjust the number of experiments while watching the histogram gradually smooth into the Gaussian distribution. The preceding example illustrates the case where two outcomes are equally probable. Many instances in nature, however, have two possible outcomes that are not equally probable. A simple variation to the program can illustrate this

bias and actually reproduce the mass spectrum of a molecule that contains two isotopes, such as Cl2. The variation is to add a case structure in place of the round to nearest integer that tests whether the randomly generated number is less than a newly added probability condition control. If it is, then the condition is “True” and a 1 is wired into the array. In Figure 5, the number of trials has also been changed to a control. If the randomly generated number is greater than the probability condition, then the condition is “False” and a zero is wired into the array. With the parameters employed earlier for the coin C




Figure 5. LabVIEW program that creates a histogram of successes for a biased system where the probability is not 50:50.

toss experiment, a probability of 0.5 will yield identical results. However, changing the probability condition will shift the histogram to the left or the right of 25 heads. The intensity of ions in a mass spectrum of Cl2 can be simulated (shown in Figure 5) by setting a condition of two trials (two chlorine atoms) and a probability condition of 0.24 (the abundance of chlorine-37).

■ ■

HAZARDS No hazards are associated with this laboratory exercise. DISCUSSION

The major components of statistical mechanics are distributed throughout a year of physical chemistry courses at the University of Mississippi. The physical chemistry laboratory course is taught in the spring (second) semester, and this laboratory exercise is normally scheduled for the first laboratory meeting. After completing this laboratory exercise, students understand how to record experimental data, the basic principles of probability, how a large number of individual Bernoulli trials and the binomial distribution yield the Gaussian distribution for a large number of measurements, how to present data, and how to create a powerful simulation using computer programming software. This laboratory exercise uses conceptually simple concepts, such as flipping a coin, to illustrate more complex ideas in statistical mechanics such as the Gaussian distribution and also introduces these concepts to students in a relatively short period of time without having to add a great deal of content to the already rigorous B.S. chemistry curriculum.43 This laboratory exercise also appeals to a variety of learners including spatial, kinesthetic, logical, interpersonal, and intrapersonal, leading to comprehension by most or all of the students in the laboratory. The student handout (Supporting Information) contains all of the Labview screenshots needed to construct the final simulation. Instructors may choose to omit this section so that students can develop their own simulation without guidance. Figure 6 shows class data accumulated over the past 6 years (2009−2014) at the University of Mississippi. The significant increase in the number of individual measurements (400 vs 80 individual trials for just the year 2014) results in a profile that

Figure 6. Coin toss data taken from laboratory classes from 2009 to 2014 (400 individual trials) compared to a Gaussian distribution calculated using the mean and standard deviation.

much better resembles the Gaussian distribution created with the distribution’s mean and standard deviation than that shown in Figure 1. Instructors may similarly build up a library of student results that can be distributed to laboratory classes each year. These accumulated results can demonstrate the effect of increasing individual experiments on the resulting experimental histogram and also serve as a comparison to simulated histograms based on the same number of individual measurements.

■

ASSOCIATED CONTENT

S Supporting Information *

Instructor notes and a sample student handout. This material is available via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. D



■


(24) Larrabee, C. E.; Sprague, E. D. Spreadsheet Programming in Introductory Physical Chemistry. J. Chem. Educ. 1988, 65, A204− A205. (25) Timmberlake, T. The Statistical Interpretation of Entropy: An Activity. Phys. Teach. 2010, 48, 516−519. (26) Algar, W. R.; Massey, M.; Krull, U. J. Assembly of a Modular Fluorimeter and Associated Software: Using LabVIEW in an Advanced Undergraduate Analytical Chemistry Laboratory. J. Chem. Educ. 2009, 86, 68−69. (27) Almaraz, R. T.; Kochis, M. Microscale Capillary Electrophoresis: A Complete Instrumentation Experiment for Chemistry Students at the Undergraduate Junior or Senior Level. J. Chem. Educ. 2003, 80, 316−319. (28) Economou, A.; Papargyris, D.; Stratis, J. Automated FlowInjection Instrument for Chemiluminescence Detection Using a LowCost Photodiode Detector: An Interdisciplinary Project in Chemical Instrumentation, Graphical Programming, Computer Interfacing, and Analytical Chemistry. J. Chem. Educ. 2004, 81, 406−410. (29) Economou, A.; Tzanavaras, P. D.; Themelis, D. G. SequentialInjection Analysis: Principles, Instrument Construction, and Demonstration by a Simple Experiment. J. Chem. Educ. 2005, 82, 1820−1822. (30) Ghiasy, D.; Tham, M. T.; Boodhoo, K. V. K. Control of a Spinning Disc Reactor: An Experimental Study. Ind. Eng. Chem. Res. 2013, 52, 16832−16841. (31) Jensen, M. B. Integrating HPLC and Electrochemistry: A LabVIEW-Based Pulsed Amperometric Detection System. J. Chem. Educ. 2002, 79, 345−348. (32) Laughlin, B. C.; Mulligan, C. C.; Cooks, R. G. Atmospheric Pressure Ionization in a Miniature Mass Spectrometer. Anal. Chem. 2005, 77, 2928−2939. (33) Lin, C.-H.; Lin, C.-H.; Li, Y.-S.; He, Y.-S. Development and Application of a Milli-Whistle for Use in Gas Chromatography Detection. Anal. Chem. 2010, 82, 7467−7471. (34) Martínez, L. M.; Videa, M.; Mederos, F.; Mesquita, J. Constructing a High-Sensitivity, Computer-Interfaced, Differential Thermal Analysis Device for Teaching and Research. J. Chem. Educ. 2007, 84, 1222−1223. (35) Muyskens, M. A.; Glass, S. V.; Wietsma, T. W.; Gray, T. M. Data Acquisition in the Chemistry Laboratory Using LabVIEW Software. J. Chem. Educ. 1996, 73, 1112−1114. (36) Ogren, P.; Nelson, S.; Henry, I. Using a Diode Laser Pointer to Count Drops and Automate Titration Systems. J. Chem. Educ. 2001, 78, 353−355. (37) Ogren, P. J.; Jones, T. P. Laboratory Interfacing Using the LabVIEW Software Package. J. Chem. Educ. 1996, 73, 1115−1116. (38) Scott, D. W.; Lane, D. M. Simulations, case studies, and an online text: A web-based resource for teaching statistics. Metrika 2000, 67−90. (39) Smith, E. T.; Hill, M. Constructing a LabVIEW-Controlled High-Performance Liquid Chromatography (HPLC) System: An Undergraduate Instrumental Methods Exercise. J. Chem. Educ. 2010, 88, 317−318. (40) Soyemi, O. O.; Busch, M. A.; Busch, K. W. Multivariate Analysis of Near-Infrared Spectra Using the G-Programming Language. J. Chem. Inf. Model. 2000, 40, 1093−1100. (41) Spanoghe, P.; Cocquyt, J.; Van der Meeren, P. A Low-Cost Dynamic Surface Tension Meter with a LabVIEW Interface and Its Usefulness in Understanding Foam Formation. J. Chem. Educ. 2001, 78, 338−342. (42) Tabert, A. M.; Griep-Raming, J.; Guymon, A. J.; Cooks, R. G. High-Throughput Miniature Cylindrical Ion Trap Array Mass Spectrometer. Anal. Chem. 2003, 75, 5656−5664. (43) Drew, S. M. Integration of National Instruments’ LabVIEW Software into the Chemistry Curriculum. J. Chem. Educ. 1996, 73, 1107−1111. (44) Garge, S. C.; Wetzel, M. D.; Ogunnaike, B. A. Inference-Based Scheme for Controlling Product End-Use Properties in Reactive Extrusion Processes. Ind. Eng. Chem. Res. 2010, 49, 8021−8034.

ACKNOWLEDGMENTS N.I.H. acknowledges support from the U.S. National Science Foundation EPS-0903787, Grant CHE-0955550, as well as the University of Mississippi College of Liberal Arts, Department of Chemistry and Biochemistry, Office of Research and Sponsored Programs, and the NSF-funded Ole Miss Physical Chemistry Summer Research Program REU (Grant CHE-1156713). We also acknowledge John Kelly, Kristina Cuellar, Louis McNamara, and Debra Jo Scardino Sage for their contributions to the development of this laboratory exercise.

■

REFERENCES

(1) Cooksy, A. Physical Chemistry: Quantum Chemistry and Molecular Interactions, 1st ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2014. (2) Cooksy, A. Physical Chemistry: Thermodynamics, Statistical Mechanics & Kinetics, 1st ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2014. (3) Engel, T.; Reid, P. J. Quantum Chemistry & Spectroscopy, 3rd ed.; Pearson: Boston, 2013. (4) Engel, T.; Reid, P. J. Thermodynamics, Statistical Thermodynamics, & Kinetics, 3rd ed.; Pearson: Boston, 2013. (5) Garrod, C. Statistical Mechanics and Thermodynamics; Oxford University Press: New York, NY, USA, 1995. (6) von Baeyer, H. C. Warmth Disperses and Time Passes: The History of Heat; The Modern Library: New York, NY, USA, 1999; p 88. (7) Dill, K. A.; Bromberg, S. Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience, 2nd ed.; Garland Science: New York, NY, USA, 2010. (8) Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 8th ed.; McGraw-Hill: New York, NY, USA, 2008. (9) Reif, F. Fundamentals of Statistical and Thermal Physics; McGrawHill: New York, NY, USA, 1965. (10) Engel, T.; Reid, P. J. Physical Chemistry, 3rd ed.; Pearson: Boston, 2013. (11) Wallner, A. S.; Brandt, K. A. The Validity of Stirling’s Approximation. J. Chem. Educ. 1999, 76, 1395−1397. (12) Barrow, G. M. Computer-Based Studies for Physical Chemistry. J. Chem. Educ. 1980, 57, 697−702. (13) Ogilvie, J. F.; Monagan, M. B. Teaching Mathematics to Chemistry Students with Symbolic Computation. J. Chem. Educ. 2007, 84, 889−896. (14) Kraska, T.; Romer, F.; Imre, A. R. The Relation of Interface Properties and Bulk Phase Stability: Molecular Dynamics Simulations of Carbon Dioxide. J. Phys. Chem. B 2009, 113, 4688−4697. (15) Teleman, O.; Jonsson, B.; Engstrom, S. LUSimA Macintosh Program for Simulation of Molecular and Polymer Systems. J. Chem. Educ. 1993, 70, 641−649. (16) Serrano, A.; Santos, F. v. M. T.; Greca, I. M. Teaching Ionic Solvation Structure with a Monte Carlo Liquid Simulation Program. J. Chem. Educ. 2004, 81, 1322−1330. (17) Maruacco, C. J. Spreadsheet Simulation of a Simple Kinetic System. J. Chem. Educ. 1993, 70, 993−994. (18) Xie, Q.; Tinker, R. Molecular Dynamics Simulations of Chemical Reactions for Use in Education. J. Chem. Educ. 2006, 83, 77−83. (19) Criado, C.; López, D. A Pedagogical Simulation of Maxwell’s Demon Paradox. J. Chem. Educ. 2004, 81, 1679. (20) Pharr, D. Y. Teaching Risk Assessment in Undergraduate Chemistry Using BCTC. J. Chem. Educ. 1993, 70, 294−295. (21) Muñoz de la Peña, A.; González-Gómez, D.; Muñoz de la Peña, D.; Gómez-Estern, F.; Sánchez Sequedo, M. Automatic Web-Based Grading System: Application in an Advanced Instrumental Analysis Chemistry Laboratory. J. Chem. Educ. 2013, 90, 308−314. (22) Starzak, M. E. Computer Examination of Some Basic Probability Models. J. Chem. Educ. 1974, 51, 717−722. (23) Elmatad, Y.; Zitolo, M.; Fera, D.; Jerschow, A. Examining Gas Kinetics in MATLAB. Chem. Educ. 2007, 12, 89−93. E




(45) Glendening, E. D.; Kansanaho, J. M. Spektri-Sim: Interactive Simulation and Analysis of the Infrared Spectra of Diatomic Molecules. J. Chem. Educ. 2001, 78, 824−826. (46) Henderson, G. A Virtual Curve-Fitting Instrument for Interactive Analysis of Excimer Dynamics. J. Chem. Educ. 1999, 76, 868−870. (47) Hilty, C.; Bowen, S. An NMR Experiment Based on Off-theShelf Digital Data-Acquisition Equipment. J. Chem. Educ. 2010, 87, 747−749. (48) Howard, A. A.; Tschumper, G. S.; Hammer, N. I. Effects of Hydrogen Bonding on Vibrational Normal Modes of Pyrimidine. J. Phys. Chem. A 2010, 114, 6803−6810. (49) Huang, Z.; Munro, N.; Hühmer, A. F. R.; Landers, J. P. AcoustoOptical Deflection-Based Laser Beam Scanning for Fluorescence Detection on Multichannel Electrophoretic Microchips. Anal. Chem. 1999, 71, 5309−5314. (50) Allerhand, A.; Dobie-Galuska, A. Implementation of LabVIEW for Computer-Controlled Experiments in General Chemistry Laboratory Instruction. Chem. Educ. 2000, 5, 71−76. (51) Belletti, A.; Borromei, R.; Ingletto, G. Teaching Physical Chemistry Experiments with a Computer Simulation by LabVIEW. J. Chem. Educ. 2006, 83, 1353−1355. (52) Belletti, A.; Borromei, R.; Ingletto, G. Determination of Ka and Kb Equilibrium Constants by pH Measurements: A Computer Simulation by LabVIEW. Chem. Educ. 2010, 15, 121−125. (53) Goodney, D. E. Using icon-based programming for classroom demonstrations. J. Chem. Educ. 1992, 69, 811−814. (54) Jensen, M. B. Using Web-Based Resources To Incorporate LabVIEW into an Instrumental Analysis Course. J. Chem. Educ. 2009, 86, 525−527. (55) Gostowski, R. Teaching Analytical Instrument Design with LabVIEW. J. Chem. Educ. 1996, 73, 1103−1107. (56) Rupright, M. E.; Smith, E. T. Processing Noisy SignalsComputer Laboratory Exercise. Chem. Educ. 2006, 11, 283−286. (57) McDonald, A. G.; Tipton, K. F. Computer-Controlled System for the Study of Oxidase Reactions: Application to the Peroxidase− Oxidase Oscillator. J. Phys. Chem. B 2010, 114, 16244−16252. (58) Hamilton, T. M.; Dobie-Galuska, A. A.; Wietstock, S. M. The oPhenylenediamine-Horseradish Peroxidase System: Enzyme Kinetics in the General Chemistry Laboratory. J. Chem. Educ. 1999, 76, 642− 644. (59) Marty, M. T.; Beussman, D. J. Simulating a Time-of-Flight Mass Spectrometer: A LabView Exercise. J. Chem. Educ. 2013, 90, 239−243. (60) Ayers, T. M.; Akin, S. T.; Dibble, C. J.; Duncan, M. A. Laser Desorption Time-of-Flight Mass Spectrometry of Inorganic Nanoclusters: An Experiment for Physical Chemistry or Advanced Instrumentation Laboratories. J. Chem. Educ. 2014, 91, 291−296. (61) Belletti, A.; Borromei, R.; Ingletto, G. The Conductivity of Strong Electrolytes: A Computer Simulation in LabVIEW. Chem. Educ. 2008, 13, 244−247. (62) Woolfolk, A. Educational Psychology, 12th ed.; Pearson: Boston, 2012.

F


Studying the Binomial Distribution Using LabVIEW - Journal of

Recommend Documents