In this modern world, mathematics is a very important subject. Social sciences and physical sciences which are all quantitative sciences are based on it. The subjects provide a theoretical background for statistics, physical and computer sciences and data analysis. Apparently mathematics and statistics are highly valued by employers, hence graduates have an exceptional range of career option. However, statistics is critical to the world economy, particularly as the age of big data grows in each and every day.
Generally, mathematics and statistics incorporate calculus, algebra fluid mechanics and also mathematical biology. Skills like decision making and problem-solving are acquired as one progresses and experience in software packages. However, these packages are used when conducting a statistical investigation and presenting the results. Statistics as a branch of mathematics entails data collection, analysis, organization, interpretation, and presentation. Additionally, it assists individuals in developing knowledge in finding patterns, analyzing large data sets, drawing conclusions and dealing with nonfigurative concepts.
Importance of Statistics
Individuals with knowledge about statistics are well equipped with a wide range of expertise in data and probability hence it’s possible for them to further their skills in applied mathematics or pure mathematics. Application of statistics to industrial or social problem can start with a model or population to be studied. Basically, mathematicians think in a logical manner and to approach a certain problem in creative and analytical ways. Statistics takes every aspect of data which includes preparation of data collection through designing of experiments and surveys. Experiments involves collection of measurements of a study being carried out, manipulating the system to determine if the values have been modified.
Generally, each and everyday individuals sharpen their mathematical skills in areas such as optimization, environmental modeling, computational mathematics, and nonlinear phenomena. Evidently, the areas elaborate on matters concerning analysis, construction, and clarification of mathematical models that describe the future market, the atmosphere, and cell growth. For instance, in statistics, there are two methods which are used in data analysis and they include inferential and descriptive approaches. Inferential statistics draws assumption from the data which is subject to random variation while descriptive summarizes from a sample using standard deviation and mean.
The goal of statistics is to do research on projects and investigate interconnection, and eventually to draw conclusions on the outcome of changes in predictor or independent variables on dependent ones. The basic procedures of a statistical experiment include planning the research, designing of experiments, performing the experiment, exploring the data to suggest the new hypothesis and finally recording and presenting the results of study.
Core Units in Mathematics and Statistics
Mathematics and statistics comprises of core units such us discrete mathematics, calculus one, probability and statistics one and mathematics for science. Discrete mathematics involves discrete elements that use arithmetic and algebra. Moreover, the units entail concepts on graphs, enumeration, block designs, partially ordered sets, set theory discrete geometry and probability theory. Calculus 1 focuses on continuous change and students learn all about differential calculus. However, differential calculus concentrates on slopes of the curve and instantaneous change. Additionally, it makes the use of an essential notion of infinite series and sequence to a well-defined limit.
Calculus is widely used in engineering, science, and economics. Infinitesimals is a method used in developing calculus by working with small quantities. The symbols (dx) and (dy) are treated as infinitesimal and their ratios are taken to be the gradient. Differential calculus explains how the derivative of a function is applied, and the process in which gradient is obtained is called differentiation. Normally, it is possible to form a new function by finding the derivative at every point.
Probability and Statistics
Probability and statistics one, is an academic discipline where probability distributions are often used. However, the unit contains sections which discusses terms like a random experiment, random variable, conditional probability, and sample space. A random experiment is a state whose result cannot be forecasted unless it is observed. The set of all probable random experiment is called sample space. Random variables are of two types namely discrete and continuous variables.
However, discrete variables take on only a countable number of different values while continuous random variable takes unlimited numbers of possible values. Basically, probability is the degree of the likelihood that an event will take place in a random experiment. Conditional probability measures the event if and only if another event has already happened. Most of the topics covered in this unit include normal distribution, independence, and test of hypothesis, conditional probability, Bayes theorem confidence interval and eventually sampling.
Furthermore, linear algebra, vector analysis, number theory, calculus two, both probability two and three and classical mechanics are also in this field of statistics. In linear algebra, geometry is presented including the definition of basic objects which include planes, lines, and rotations. Besides, functional analysis is also applied in algebra to spaces of functions. This unit is fundamental in engineering and most sciences since it deals with modeling natural phenomena and how such models are computed. Systems of linear equations are also learned and how matrix theory is developed to solve these set of equations.
Vector Analysis
Vector analysis is also another important unit which describes quantities that has both magnitude and direction. Scalars are geometric quantities can be defined by stating their magnitude in units of measures. A vector quantity is characterized by a line segment with an arrow pointing in the direction of the vector. However, the length of the segment signifies the magnitude of the vector. To differentiate between a scalar and a vector, boldface letters are used to denote vectors.
Number theory is another essential tool in mathematics. It deals primarily in the study of integers especially prime numbers as well as their properties. One is able to understand questions in this unit through the study of analytical objects. Computer technology has really enabled people to make advancements in factoring large numbers, obtaining prime, solving numerical questions and testing conjectures.
Classical Mechanics
Classical mechanics is a branch of mathematics that describes the motion of objects and the forces that affect them. However, learners are taught that is the state of the object is known at a particular time, it is possible to predict its future movement and how it had moved in the past. Mathematicians like Isaac Newton mathematically developed concepts which describe the motion of bodies when a force is applied. When studying large objects, accurate results are found when the speeds of these objects are not approaching the speed of light.
Normally, other important subjects in statistics includes real analysis one, ordinary differential equations one, complex analysis one and theory of estimation. Additionally, hypothesis testing, operations research one, and numerical analysis are also vital subjects in mathematics and statistics. Each and every unit is crucial hence for instance, real analysis studies the behaviors of sequences, series, and real numbers. Mainly, more about limits, convergence, smoothness, continuity, integration, and differentiation are well explained in this unit. The course covers abstract concepts and elaborates on construction and understanding of proofs.
Numerical Analysis
The numerical analysis involves the study of processes that use numerical estimates for mathematical analysis problems. Evidently, numerical analysis is applied in all fields of physical sciences, engineering, medicine, social sciences and even businesses. Before computers were introduced, numerical approaches depended on hand interpolation formulas. In general, numerical analysis elaborates on algorithms which use methods that depend on interpolation formulas functional to data from a large printed table.
The study of errors also forms a vital part of numerical analysis. The field of numerical analysis involves various disciplines which include computing values of a function, solving Eigen values, and extrapolation, interpolation and regression. Apparently, the goal of this unit is the analysis and design of techniques to provide estimated but correct solutions to difficult problems.
Complex Analysis
Unlike the real analysis, complex analysis deals mainly with the study of complex numbers and their functions. The essential result of complex analysis is the Cauchy integral theorem which explains the conditions a function must satisfy for a complex derivative to exist. The course is useful in algebraic geometry, applied mathematics, and analytic combinations. Additionally, the unit is applicable in engineering fields such as aerospace, nuclear, electrical and mechanical. A complex function has a subset of complex numbers known as domain and a codomain. However, holomorphic are complex functions which are differentiable at an open subset of the complex plane.
The Theory of Estimation
The theory of estimation gives emphasis on how parameters are estimated based on empirical data measures that have a random component. Unknown parameters are approximated by an estimator using the measurements. Ultimately, the probabilistic approaches assumes that the data which is measured with probability distribution depends on the parameter of interest. Also, the measured data vector with reference to the approach of set membership fits the set which depends on the parameter vector. Some of the commonly used topics in this unit include maximum likelihood estimator, Cramer Rao bound, least squares and minimum variance unbiased estimator.
Ordinary Differential Equations
Ordinary differential equations (ODEs) are differential equations comprising of at least one function containing one independent variable and the slope of those functions. This unit emerges from many contexts of mathematics, natural and social sciences. Gradients and differentials are applied in the description of the mathematical change. A number of derivatives, differentials, and functions become connected through equations, such that differential equation describes dynamically changing variation, phenomena, and evolution. ODEs are generally applied to scientific fields including astronomy, physics, chemistry, and economics among others.
Test Hypothesis
Test of hypothesis explains the testable hypothesis on the basis of process observation that is modeled with respect to random variables. However, this test is a method of statistical inference. From an idealized model, two statistical data sets are compared against a synthetic data set. Moreover, the relationship between these two data sets is determined by proposing a hypothesis and eventually compared as either null or alternative hypothesis. Generally, a hypothesis test is applicable when determining what result of a study would lead to rejecting the null hypothesis at a certain level of significance.
Operations Research
Operations research is an important unit in mathematics and statistics since it deals with the application of analytical and mathematical methods to the analysis and study of complex problems hence making proper decisions. Furthermore, the techniques in this course are applied and quality solutions are arrived at decisions which seems complex. In addition, the discipline is highly employed in industrial engineering, psychology, and operations management. It is used to form strategies which are of high level, improve day to day operations and design public policies.
Eventually, mathematics and statistics also entail disciplines such as partial differential equation, design, and analysis of experiments, time series analysis, and design and analysis of sample survey. Furthermore, it comprises of regression modeling, quality control methods, non-parametric methods, stochastic processes, multivariate analysis and simulation, and modeling.
Time Series
Time series analysis represents a series of data points that shows how sequences are taken at successive equally spaced points at a certain period of time. A perfect example of time series are heights of ocean tides. However, they are plotted frequently using line charts. Moreover, these series are apparently used in signal processing, econometrics, statistics, predicting earth tremors and weather forecasting.
Consequently, time series are methods for analyzing data so as to produce a meaningful characteristic of the data and other statistics. On the other hand, the multivariate analysis comprises of observing and analyzing multiple statistical variable results at a time. The analysis is used in designing capabilities, inverse design, analysis of alternatives and determination of correlation across hierarchical levels.
Non-Parametric Method
Nonparametric methods is also a discipline in statistics that do not need assumptions from the population being analyzed. However, the method does not depend on the population of interest. Moreover, the method is also referred to as distribution-free method since the set of parameters are not fixed nor are they distributed. Examples of these methods learned include Spearman correlation test, sign test for a population mean and two independent means U test.
Regression Modeling
Basically, regression modeling comprises of the theory explaining how the relationship among variables is estimated through application of statistical processes. More importantly, regression analysis is used in time series modeling and estimation. In prediction and forecasting, regression analysis is widely applied and also when studying which independent variables are related to the dependent variables. Also, it allows the comparison of variables measured on diverse scales such as the effect of price variations. Parameters which are involved in the model include the unknown parameter denoted as Beta, the predictor variable X, and the response variable Y.
Design and Analysis of Experiments
Design and analysis of experiments is a program that examines how experiments are designed, carried out and how the data obtained is analyzed. A number of designs are discussed and their differences, merits, and demerits are also noted. Moreover, fractional factorial is also elaborated deeply and with clarity. In this section, experiments are aimed at forecasting the results by introducing a change in the input variable. The design not only involves selection of appropriate dependent, independent and control variables but preparing the delivery of experiment under conditions which are statistically optimal.
Stochastic Process
A stochastic process is also among the mathematics disciplines which is defined as a random process that include a collection of associated random variables. These processes are applied in mathematical models that appear to differ in a random manner. However, stochastic processes are useful in sciences such as chemistry, biology, physics, and neuroscience. These process can be categorized in different ways such as by the state of space and the dependence between the random variables. One of the ways of classifying is by the cardinality of the state space and index set.
Conclusion
In conclusion, when a student pursues a Bachelor of Science in mathematics and statistics, one can major in actuarial science and masters of Science in pure and applied mathematics. Where business is becoming more oriented to data, this course is much applied since students are well-founded with skills of analyzing data and offering advice on risk and financial management accordingly.
