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• If is increased while and are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location. • If is decreased while and are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or ), and its height increases.

Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter,, and the scale parameter,, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where or the 1-parameter form where constant, can easily be made. Effects of the Shape Parameter, beta The Weibull shape parameter,, is also known as the slope. This is because the value of is equal to the slope of the regressed line in a probability plot.

This is always at 63.2% since: Now any reliability value for any mission time can be obtained. For example, the reliability for a mission of 15 hours, or any other time, can now be obtained either from the plot or analytically. To obtain the value from the plot, draw a vertical line from the abscissa, at hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read, in this case. This can also be obtained analytically from the Weibull reliability function since the estimates of both of the parameters are known or: Probability Plotting for the Location Parameter, Gamma The third parameter of the Weibull distribution is utilized when the data do not fall on a straight line, but fall on either a concave up or down curve. The following statements can be made regarding the value of: • Case 1: If the curve for MR versus is concave down and the curve for MR versus is concave up, then there exists a such that, or has a positive value. • Case 2: If the curves for MR versus and MR versus are both concave up, then there exists a negative which will straighten out the curve of MR versus.

The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see ), is the straight line: The corresponding equations for and are: and: where: and: and the values are again obtained from the median ranks. Once and are obtained, solve the linear equation for, which corresponds to: Solving for the parameters from above equations, we get: and The correlation coefficient is evaluated as before. RRX Example Again using the same data set from the and examples (with six failures at 16, 34, 53, 75, 93 and 120 hours), calculate the parameters using rank regression on X. Solution The same table constructed above for the can also be applied for RRX. Using the values from this table we get: or: and: or: Therefore: and: The correlation coefficient is: The results and the associated graph using Weibull++ are shown next. Note that the slight variation in the results is due to the number of significant figures used in the estimation of the median ranks.

Reliasoft Weibull Training

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Using these non-informative prior distributions, can be rewritten as: The one-sided upper bounds of is: The one-sided lower bounds of is: The two-sided bounds of is: Same method is used to obtain the bounds of. Bounds on Reliability From the posterior distribution of we have: The above equation is solved numerically for. The same method can be used to calculate the one sided lower bounds and two-sided bounds on reliability. Bounds on Time From, we know that: From the posterior distribution of, we have: The above equation is solved numerically for. The same method can be applied to calculate one sided lower bounds and two-sided bounds on time.

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Course overview - 3 days Weibull and Life Data Analysis provides the foundational knowledge for all aspects of reliability engineering education. The fundamental teachings of this course are an important component of an effective, comprehensive reliability program, ensuring that reliability professionals are proficient in the concepts of reliability engineering mathematics and basic reliability data analysis. Software used This course will help you understand how life data analysis methodologies can be applied using with practical hands-on exercises and case study examples.

The software is also packed with tools for related reliability analyses, such as warranty data analysis, degradation data analysis, non-parametric data analysis, recurrent event data analysis, reliability test design and experiment design & analysis (DOE). • Supports all types of life data, including complete, right censored (suspended), left censored, interval censored and free-form data, entered individually or in groups. • Includes all major lifetime distributions (including the Weibull model and the mixed Weibull model as well as the Exponential, Lognormal, Normal, Generalized Gamma, Gamma, Logistic, Loglogistic, Gumbel, Bayesian-Weibull and Competing Failure Modes). A distribution wizard will help you select the one with the best fit for a particular data set. • Flexible results, plots and reports to quickly provide the results, plots and reports you need to support decisions based on analysis results and two template-based reporting tools that incorporate results based on the data analysis along with your own custom calculations. The software provides a choice of all major lifetime distributions (including all forms of the Weibull distribution).

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• Quick Parameter Estimator: Estimates the parameters of a statistical model based on what you know about the behavior over time. The software will 'translate' your answers into the inputs required to obtain the model parameters. • Quick Statistical Reference: Frees you from tedious lookups in tables by quickly returning results for commonly used statistical functions.

• Home page: • Last updated: December 26th, 2011 Reliasoft weibull 7 torrent in Description.

This gives you the flexibility to perform simple parameter estimation on statistical models other than the life distributions and life-stress relationship models available in standard folios. After you have solved the equation, the tool provides a plot to visualize how the data fit the function, and also makes it easy to calculate Y for any given X value. • Non-Linear Equation Root Finder: Quickly solves for the root of any user-defined non-linear equation. This helps you eliminate some of the guesswork of solving for the value of the unknown variable that makes the function be as close to zero as possible. Casper windows 10. • Quick Parameter Estimator: Estimates the parameters of a statistical model based on what you know about the behavior over time.

• Home page: • Last updated: December 26th, 2011 Reliasoft weibull 7 torrent in Description.

Course overview - 3 days Weibull and Life Data Analysis provides the foundational knowledge for all aspects of reliability engineering education. The fundamental teachings of this course are an important component of an effective, comprehensive reliability program, ensuring that reliability professionals are proficient in the concepts of reliability engineering mathematics and basic reliability data analysis.

ReliaSoft Weibull++ is the industry standard in for thousands of companies worldwide. Weibull++ provides the most comprehensive toolset available for reliability life data analysis, calculated results, plots and reporting. The software supports all data types and all commonly used product lifetime distributions. The software is also packed with tools for related reliability analyses, such as warranty data analysis, degradation data analysis, non-parametric data analysis, recurrent event data analysis, reliability test design and experiment design & analysis (DOE). • Supports all types of life data, including complete, right censored (suspended), left censored, interval censored and free-form data, entered individually or in groups.

• The parameter may assume all values and provides an estimate of the earliest time a failure may be observed. A negative may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.

Least Squares Analysis 1 16 2.7726 0.1091 -2.1583 7.6873 4.6582 -5.9840 2 34 3.5264 0.2645 -1.18 1.393 -4.1620 3 53 3.9703 0.4214 -0.60 0.3637 -2.3943 4 75 4.3175 0.5786 -0.1 0.0213 -0.6303 5 93 4.5326 0.7355 0.28 0.0813 1.2923 6 120 4.7875 0.8909 0.79 0.6328 3.80 -3.0 7.1502 -8.0699 Utilizing the values from the table, calculate and using the following equations: or: and: or: Therefore: and: or: The correlation coefficient can be estimated as: This example can be repeated in the Weibull++ software. The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds). Rank Regression on X Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the horizontal deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring the reliability function into a linear form. This step is exactly the same as in the regression on Y analysis and all the equations apply in this case too.