Application of Probability and Statistics in Science
Probability is an area of study which involves predicting the relative likelihood of various outcomes. It is a mathematical area that has developed over the past three or four centuries. One of the early uses was to calculate the odds of various gambling games. Its usefulness for describing errors of scientific and engineering measurements was soon realized. Engineers study probability for its many practical uses, ranging from quality control and quality assurance to communication theory in electrical engineering. Engineering measurements are often analyzed using statistics, Â and a good knowledge of probability is needed in order to understand statistics.
- Research Design
The use of statisticalÂ Â and probability tests inÂ Â is an important aspect of science that adds strength and certainty to scientific conclusions. For example, in 1843,Â , an English entrepreneur, founded the Rothamsted Experimental Station in Hertfordshire, England to investigate the impact of fertilizer application on crop yield. Lawes was motivated to do so because he had established one of the first artificial fertilizer factories a year earlier. For the next 80 years, researchers at the Station conductedÂ Â in which they applied fertilizers, planted different crops, kept track of the amount of rain that fell, and measured the size of the harvest at the end of each growing season.Â
By the turn of the century, the Station had a vast collection ofÂ Â but few useful conclusions: One fertilizer would outperform another one year but underperform the next, certain fertilizers appeared to affect only certain crops, and the differing amounts of rainfall that fell each year continually confounded theÂ Â (Salsburg, 2001). The data were essentially useless because there were a large number of uncontrolledÂ .Â
- Computer Science
Probability and Statistics for Computer Science develops introductory topics in probability with this particular emphasis, providing computer science students with an invaluable resource in their continued studies and professional research.Â
Probability and Statistics for Computer Science treat the most common discrete and continuous distributions, showing how they find use in decision and estimation problems, and constructs computer algorithms for generating observations from the various distributions. This one-of-a-kind resource also: Includes a thorough and rigorous development of all the necessary supporting mathematics Provides an opportunity to reconnect applications with the theoretical concepts of distributions introduced in prerequisite courses Gathers supporting topics in an appendix: set theory, limit processes, real number structure, Riemann-Stieltjes integrals, matrix transformation, and determinants Uses computer science examples such as client-server performance evaluation and image processing The author also addresses a variety of supporting topics, such as estimation arguments with limits, properties of power series, and Markov processes. Johnson's text proves an ideal resource for computer science students and practitioners interested in a probability study specific to their field.Â
- Environmental Engineering
Issues related to environmental risk assessment include health effects, impact on natural resourcesÂ or man-made structures due to pollution, change in climatic conditions, water quality of streams, etc. There are different parametric, non-parametric and empirical models are used to address theseÂ issues. Probability methods play a role in:
- Estimation of model parameters.
- Identification ofÂ probability distribution.
- Determination of dependencies among variables.
- Estimation of model uncertainties etc.
- Geotechnical Engineering
- Hydrology and Water Resources Engineering
Uncertainties in hydrology and water resources engineering arise from the incompleteness of historical data, limitations in the adequate representation of sample data, the variability of hydrologic data, uncertain predictions, etc. Assessment of uncertainty is carried out through different probability methods, i.e., distribution fitting to data, probability, and quantile estimation, interval estimation of parameters, etc.
Hydrologic extreme events, such as Probable Maximum Precipitation (PMP) and Probable Maximum Flood (PMF) are estimated from historical data. However, the estimation procedure requires different probability methods. Different hydrologic variables, such as rainfall, streamflow are inherently uncertain. Prediction of such hydrologic variables is often required a probabilistic method to quantify the uncertainty associated with the predictions. In the design of hydraulic structures, estimation of design life and risk analysis also requires probability methods.
- Structural EngineeringÂ Â
In structural engineering, failure can cause excessive monetary loss and injury or death. Thus, an extremely low rate of failure is assured in design. Safety factors are determined by considering the risk or probability of failure. The concept of low-probability high-consequence risk events is the key issue in the design of complex structures, such as offshore structures, nuclear plants, and high-exposure public structures. Sources of uncertainty in structural engineering lie in the magnitude of the load, the strength of structural material, the number of load cycles until fatigue failure, etc.