Mathematics A3 for Chemical Engineers and Bioengineers

Experiments, outcomes, events and sample space.  Discrete and continuous sample space. Operation on the set of events, properties of the operations, De Morgan’s laws. Mutually exclusive, impossible and certain events. Relative frequency,  probability function, axioms and basic properties of probability. Conditional probability. Multiplication rule. Partition of the sample space. The law of total probability and Bayes’ rule. Independence of two or more events. Independent experiments.

Finite sample space and enumaration. Combinatorial analysis, permutations, combinations, multinomial coeeficients. Sample spaces having equally likely outcomes.

One-dimensional random variables. Distribution function, and basic properties. Discrete, continuous and mixed random variables. Discrete random variables, the probability mass function. Continuous random variabales, probabilty density function, and properties. Some parameters of random variables: expectation, variance, median and mode of random variables, basic properties.

Some important discrete random variables: the Bernoulli, the binomial, the geometric, the uniform and Poisson distributions, and their applications. Poisson process. Some important  continuous random variables: the uniform, the exponential, the normal distrubutions and their applications. The Moivre-Laplace limit theorem.

Markov’s and Chebyshev’s inequalities, the law of large numbers, applications.  Computations with random variables, transforming discrete and continuous random variables.

Jointly distributed random variables. Joint distribution functions, basic propetries.  Marginal distributions.   Joint probability density functions, properties. Independent random variables. Independence of random variables, continuous and discrete cases. Degree of association between two variables,the covarianace and the correlation coefficient of two random variables, properties. Conditional distributions, conditional expectation, regression of the mean. The bivariate normal distribution.

The central limit theorem, applications.  

Introduction to statistics, descriptive and inferential statistics, scale of measurment.   Graphing quntitative and qualitatve data. Parameter estimation, properties of estimatiors, biased and unbiased estimators of parameters. Estimate of mean and variance.

Confidence interval estimation ont the mean, variance known. Confidence level and precision of estimation, choice of a sample size. Confidence interval on the mean, variance unknown.

Tests of hypotheses, type I and II errors.   One-sided and two-sided hypotheses.  Test of hypotheses on the mean, variance known, Simple linear regression and correlation.

Basic concepts and ideas of differentila equations, initial value problem (review)  Exact defferential equations, integrating factors depending only on one variable. Numerical methods for an initial value problem. Euler method. Successive approximation method.