Engineering statistics probability distributions

In this chapter of statistics for engineers we have so far introduced four important probability distributions used in statistical tests and estimates. These are ... 

The phrase degrees of freedom is not interpreted identically when it is used in the different branches of science. In physics and chemistry, each independent mode in which a particle or system may move or be oriented is one degree of freedom. In mechanical engineering, degrees of freedom describes flexibility of motion. In statistics, the degrees of freedom are the number of parameter values in probability distributions that are free to be varied. In statistical mechanics the number of degrees of freedom a given system has is equal to the minimum number of independent parameters necessary to uniquely determine the location and orientation of the system in physical space. 

Z. P. Bazant, Probability distribution of energetic-statistical size effect in quasibrittle Probabilistic Engineering Mechanics, 19, 307-319, (2004). 

As stated above, use of queuing and simulation models requires collection of data regarding the probability distribution of interarrival times for customers and service times. Use of a specific queuing model requires that the assumptions regarding the probability distributions in the model are vaMd. Industrial engineers must be able to verify that the collected data fit the assumed probability distribution. Similarly, simulation models need the data regarding various service times and other probabilistic elements of the model. Statistical methods have also been used to study and manage variability in demand for services (Sahney 1982). 

FIGURE 4.2.1 Various probability distributions important in biology. The normal distribution is used for most applications. The t-distribution is used for small sample sizes from a normal distribution. The log normal distribution fits some data better than a normal distribution. The F distribution is used to check equality of variances, and the yj- (chi square) distribution is used to check expected values of data. The curves shown here are for various values of distribution parameters. (From Barnes, J.W., Statistical Analysis for Engineers and Scientists A Computer-Based Approach, McGraw-Hill, New York, 1994.)... 

Step 4 According to the central limit theorem of statistics, the probability distribution of MCST is an approximate normal distribution. The peak error of this distribution at a one-sided 95% confidence level is evaluated by (2.35), which is the engineering uncertainty of the Super LWR ... 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

FIG. 8-47 Probabilities associated with the normal distribution. (Source Montgomery and Runger, Applied Statistics and Probability for Engineers, 3d ed., Wiley, New York, 2002.)... 

A probability density function that is an approximation to the biomodal distribution and is characterized by its mean being equal to its variance. See Mezei, L.M., Practical Spreadsheet Statistics and Curve Fitting for Scientists and Engineers, Prentice-Hall, Englewood Cliffs, NJ, 1990 Dowdy, S.M. and Wearden, S., Statistics for Research, Wiley, New York, 1991 Balakrishnan, N. and Nevzorov, V.B., A Primer on Statistical Distributions, Wiley, Hoboken, NJ, 2003. 

To illustrate jointly distributed r.v. s, we consider data on grades (A = 4, B = 3, C = 2, D = l)ina probability and a statistics course for a sample of 200 Northwestern University undergraduate engineering smdents. A tabulation of these grades is shown in Table 2.5. Suppose the transcript of one of these smdents is drawn at random. What is the probability that the student received an A in probability and a B in statistics ... 

WeibuU distribution (WeibuU 1961) is probably one of the most widely used distributions in lifetesting applications. One of the reasons is the ease with which graphic procedures can be used to estimate the parameters of the WeibuU distribution and thus the reliabihty of the product. Confidence Umits can also be easUy developed. In addition, various statistical estimation procedures have recently been developed, and these can also be easily used by reUabUity engineers (Nelson 1990 Abemethy 1996 Kapur and Lamberson 1977). 

Use your basic reference books on mathematics and vector analysis to support the reading on molar mass distributions. Some examples are the (I960-) International Dictionary of Applied Mathematics. Van Nostrand, Princeton, NJ Feller W (1950-) An Introduction to Probability Theory and Its Applications. Wiley, New York Mood AM (1950) Introduction to the Theory of Statistics. McGraw-Hill, New York Hamming RW (1962-) Numerical Methods for Scientists and Engineers. Dover, New York. 

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