Take A Pdf And A Defined Function Calculate Expected Value Anf Variance In Matlab
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- Beta distribution
- Expected Values for Continuous Random Variables
- The Weibull Distribution
Documentation Help Center. The normal distribution, sometimes called the Gaussian distribution, is a two-parameter family of curves. The usual justification for using the normal distribution for modeling is the Central Limit theorem, which states roughly that the sum of independent samples from any distribution with finite mean and variance converges to the normal distribution as the sample size goes to infinity.
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Chapter 8: The Weibull Distribution. Generate Reference Book: File may be more up-to-date. The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. The advantage of doing this is that data sets with few or no failures can be analyzed.
Recalling that the reliability function of a distribution is simply one minus the cdf , the reliability function for the 3-parameter Weibull distribution is then given by:. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully. The Weibull distribution is widely used in reliability and life data analysis due to its versatility.
Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors.
Note that in the rest of this section we will assume the most general form of the Weibull distribution, i. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. It is easy to see why this parameter is sometimes referred to as the slope.
This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.
The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, using either least squares rank regression or maximum likelihood estimation MLE. One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from Kececioglu .
Assume that six identical units are being reliability tested at the same application and operation stress levels. All of these units fail during the test after operating the following number of hours: 93, 34, 16, , 53 and Estimate the values of the parameters for a 2-parameter Weibull distribution and determine the reliability of the units at a time of 15 hours. The steps for determining the parameters of the Weibull representing the data, using probability plotting, are outlined in the following instructions.
First, rank the times-to-failure in ascending order as shown next. Obtain their median rank plotting positions. Median ranks can be found tabulated in many reliability books. They can also be estimated using the following equation:. The times-to-failure, with their corresponding median ranks, are shown next. On a Weibull probability paper, plot the times and their corresponding ranks. A sample of a Weibull probability paper is given in the following figure.
The points of the data in the example are shown in the figure below. Draw the best possible straight line through these points, as shown below, then obtain the slope of this line by drawing a line, parallel to the one just obtained, through the slope indicator. Draw a vertical line through this intersection until it crosses the abscissa.
This is always at 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. This can also be obtained analytically from the Weibull reliability function since the estimates of both of the parameters are known or:.
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 other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it.
Performing rank regression on Y requires that a straight line mathematically be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized. This is in essence the same methodology as the probability plotting method, except that we use the principle of least squares to determine the line through the points, as opposed to just eyeballing it.
The first step is to bring our function into a linear form. For the two-parameter Weibull distribution, the cumulative density function is:. The least squares parameter estimation method also known as regression analysis was discussed in Parameter Estimation , and the following equations for regression on Y were derived:. Consider the same data set from the probability plotting example given above with six failures at 16, 34, 53, 75, 93 and hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the 2-parameter Weibull distribution.
From this point on, different results, reports and plots can be obtained. 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. 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 Parameter Estimation , is the straight line:.
Again using the same data set from the probability plotting and RRY examples with six failures at 16, 34, 53, 75, 93 and hours , calculate the parameters using rank regression on X.
Note that the slight variation in the results is due to the number of significant figures used in the estimation of the median ranks. The goal in this case is to fit a curve, instead of a line, through the data points using nonlinear regression. Then the nonlinear model is approximated with linear terms and ordinary least squares are employed to estimate the parameters. This procedure is iterated until a satisfactory solution is reached.
Note that other shapes, particularly S shapes, might suggest the existence of more than one population. In these cases, the multiple population mixed Weibull distribution , may be more appropriate.
The results and the associated graph for the previous example using the 3-parameter Weibull case are shown next:. As outlined in Parameter Estimation , maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function.
This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution.
Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates.
The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in Appendix D.
One last time, use the same data set from the probability plotting , RRY and RRX examples with six failures at 16, 34, 53, 75, 93 and hours and calculate the parameters using MLE. In this case, we have non-grouped data with no suspensions or intervals, i. The equations for the partial derivatives of the log-likelihood function are derived in an appendix and given next:.
Note that the decimal accuracy displayed and used is based on your individual Application Setup. The biasness will affect the accuracy of reliability prediction, especially when the number of failures are small. The software will use the above equations only when there are more than two failures in the data set. One of the methods used by the application in estimating the different types of confidence bounds for Weibull data, the Fisher matrix method, is presented in this section.
The complete derivations were presented in detail for a general function in Confidence Bounds. One of the properties of maximum likelihood estimators is that they are asymptotically normal, meaning that for large samples they are normally distributed.
The lower and upper bounds on the parameters are estimated from Nelson  :. Note that the variance and covariance of the parameters are obtained from the inverse Fisher information matrix as described in this section. The local Fisher information matrix is obtained from the second partials of the likelihood function, by substituting the solved parameter estimates into the particular functions.
This method is based on maximum likelihood theory and is derived from the fact that the parameter estimates were computed using maximum likelihood estimation methods. When one uses least squares or regression analysis for the parameter estimates, this methodology is theoretically then not applicable. However, if one assumes that the variance and covariance of the parameters will be similar One also assumes similar properties for both estimators. This gives consistent confidence bounds regardless of the underlying method of solution, i.
This is an indication that these assumptions were violated. The bounds on reliability can easily be derived by first looking at the general extreme value distribution EVD. Its reliability function is given by:. Using the equations derived in Confidence Bounds , the bounds on are then estimated from Nelson  :. This means that one must be cautious when obtaining confidence bounds from the plot.
The bounds around the time estimate or reliable life estimate, for a given Weibull percentile unreliability , are estimated by first solving the reliability equation with respect to time, as discussed in Lloyd and Lipow  and in Nelson  :. The likelihood ratio equation used to solve for bounds on time Type 1 is:. Bayesian Bounds use non-informative prior distributions for both parameters.
The same method can be used to calculate the one sided lower bounds and two-sided bounds on reliability. From Confidence Bounds , we know that:. The same method can be applied to calculate one sided lower bounds and two-sided bounds on time. The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation.
There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data.
At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions.
A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say and you would be right. The procedure of performing a Bayesian-Weibull analysis is as follows:.
The probability mass function or pmf, for short is a mapping, that takes all the possible discrete values a random variable could take on, and maps them to their probabilities. The pmf for X would be:. If we're only working with one random variable, the subscript X is often left out, so we write the pmf as p x. The mean mu or expected value E[X] of a random variable X is the sum of the weighted possible values for X ; weighted, that is, by their respective probabilities. If S is the set of all possible values for X , then the formula for the mean is:.
Expected Values for Continuous Random Variables
The generalization to multiple variables is called a Dirichlet distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. In Bayesian inference , the beta distribution is the conjugate prior probability distribution for the Bernoulli , binomial , negative binomial and geometric distributions.
This module provides functions for calculating mathematical statistics of numeric Real -valued data. The module is not intended to be a competitor to third-party libraries such as NumPy , SciPy , or proprietary full-featured statistics packages aimed at professional statisticians such as Minitab, SAS and Matlab. It is aimed at the level of graphing and scientific calculators.
Documentation Help Center. Probability distributions are theoretical distributions based on assumptions about a source population. The distributions assign probability to the event that a random variable has a specific, discrete value, or falls within a specified range of continuous values. Use Probability Distribution Objects to fit a probability distribution object to sample data, or to create a probability distribution object with specified parameter values.
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The Weibull Distribution
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