[...]... probability of 0.5 to either event Generally the probabilities of all possible events are chosen to total 1.0 If we toss two coins, we note that the fall of each coin is independent of the other The probability of either coin landing heads is thus still 0.5 The probability of both coins falling heads is the product of the probabilities of the single events, since the single events are independent:... outcomes of one kind Heads or tails with a coin nÂMðNÀMÞðNÀnÞ M objects of one kind, 2 N ðNÀ1Þ N objects of another kind k objects of kind M found in a drawing of n objects The n objects are drawn from the population without replacement after each drawing 1Àp Number of failures 2 p before the first success in a sequence of Bernoulli trials kt Random occurrence with time Probability of k occurrences in interval... specific values in a continuous distribution is: b P ða ð bÞ ¼ f ðx Þdx X (1.28) a where f(x) is the probability density function of the underlying population model Since all values of X lie between minus infinity and plus infinity ½À1; þ1Š, the probability of finding X within these limits is 1 Hence for all continuous distributions: þ1 ð f ðX Þdx ¼ 1 (1.29) À1 The expected value of a continuous distribution... usually an infinite gathering of elements-units For example, we can take each hundredth sample from a steady process and expose it to chemical analysis or some other treatment in order to establish a certain property (taking a sample from a chemical reactor with the idea of establishing the yield of chemical reaction, taking a sample out of a rocket propellant with the idea of establishing mechanical... by the middle of the nineteenth century Probably the first instance of applied statistics came in the application of probability theory to games of chance Even today, probability theorists frequently choose I Introduction to Statistics for Engineers a coin or a deck of cards as their experimental model Application of statistics in biology developed in England in the latter half of the nineteenth century... value in a definite range Examples of continuous random variables: waiting time for a bus, time between emission of particles in radioactive decay, etc The simplest probability model Probability theory was originally developed to predict outcomes of games of chance Hence we might start with the simplest game of chance: a single coin We intuitively conclude that the chance of the coin coming up heads... theory are being applied in all fields of engineering With the development of electronic computers, statistical methods began to thrive and take an ever more important role in empirical researches and system optimization Statistical methods of researching phenomena can be divided into two basic groups The first one includes methods of recording and processing-description of variables of observed phenomena... strictly determined, i.e random and deterministic phenomena are the left and right limits of stochastic phenomena In order to find stochastic relationships the present-day engineering practice uses, apart from others, experiment and statistical calculation of obtained results Statistics, the science of description and interpretation of numerical data, began in its most rudimentary form in the census... expected value of X; By appropriate manipulation, it is possible to determine the expected value of various functions of X, which is the subject of probability theory For example, the expected value of X is simply the sum of squares of the values, each weighted by the probability of obtaining the value The population variance of the random variable X is defined as the expected value of the square of the difference... not certain of having noticed all the variables At this level we do not know the natural law, i.e the mathematical model by which these factors act In this case we use experiment (empirical research) in order to reach the noticed natural law ˇ ´ Design of Experiments in Chemical Engineering Zivorad R Lazic Copyright  2004 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-31142-4 2 I Introduction . R. Lazic ´ Design of Experiments in Chemical Engineering Design of Experiments in Chemical Engineering. Z ˇ ivorad R. Lazic ´ Copyright  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:. Correlation in Linear Regression 148 1.7.2 Correlation in Multiple Linear Regression 152 II Design and Analysis of Experiments 157 2.0 Introduction to Design of Experiments (DOE) 157 2.1 Preliminary. of statistics in biology developed in England in the latter half of the nineteenth century. The first important application of statistics in the chemical industry also occurred in a factory in