Solution The first step is to calculate the likelihood function for the parameter estimates: We can now rearrange the likelihod ratio equation to the form: It now remains to find the values of and which satisfy this equation. This is an iterative process that requires setting the value of and finding the appropriate values ofand vice versa.
The upper and lower bounds on reliability are: Bounds on Time The bounds around time for a given normal percentile unreliability are estimated by first solving the reliability equation with respect to time, as follows: The next step is to calculate the variance of or: The upper and lower bounds are then found by: Likelihood Ratio Confidence Bounds As covered in Confidence Boundsthe likelihood confidence bounds are calculated by finding values for and This equation can be rewritten as: For complete data, the likelihood formula for the normal distribution is given by: For a given value ofvalues for and can be found which represent the maximum and minimum values that satisfy the above likelihood ratio equation.
These represent the confidence bounds for the parameters at a confidence levelwhere for one-sided. LR Bounds on Parameters Five units are put on a reliability test and experience failures at 12, 24, 28, 34, and 46 hours.
Solution The first step is to calculate the likelihood function for the parameter estimates: We can now rearrange the likelihood ratio equation to the form: It now remains to find the values of and which satisfy this equation.
This is an iterative process that requires setting the value of and finding the appropriate values ofand vice versa. The following table gives the values of based on given values of. This data set is represented graphically in the following contour plot: Note that this plot is generated with degrees of freedomas we are only determining bounds on one parameter.
As can be determined from the table, the lowest calculated value for is 7. Since solutions for the equation do not exist for values of below 22 or above This can be accomplished by substituting a form of the normal reliability equation into the likelihood function.
The normal reliability equation can be written as: This can be rearranged to the form: This equation can now be substituted into the likelihood ratio equation to produce an equation in terms of and: The unknown parameter depends on what type of bounds are being determined.
If one is trying to determine the bounds on time for a given reliability, then is a known constant and is the unknown parameter. Conversely, if one is trying to determine the bounds on reliability for a given time, then is a known constant and is the unknown parameter. The likelihood ratio equation can be used to solve the values of interest.
The ML estimate for the time at is This is accomplished by substituting and into the likelihood ratio equation for the normal distribution, and varying until the maximum and minimum values of are found. As can be determined from the table, the lowest calculated value for is The ML estimate for the reliability at is This is accomplished by substituting and into the likelihood ratio equation for normal distribution, and varying until the maximum and minimum values of are found.The Bivariate Normal Distribution This is Section of the 1st edition () of the book Introduc-tion to Probability, by D.
P. Bertsekas and J. N. Tsitsiklis. The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature.
Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. A normal distribution is a symmetric distribution which is centered at the mean value and the width of the distribution depends on the standard deviation.
For the normal distribution, the most frequently occurring values are close to the mean of the data set. This content was STOLEN from schwenkreis.com - View the original, and get the already-completed solution here! Discuss the concept of the normal distribution, why it is important, and what you think it .
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If the data follow a continuous and multivariate normal distribution, then maximum likelihood attains optimal asymptotic properties, viz., that the estimates are normal, unbiased, and efficient. The effects of non-normality on estimates, standard errors, and tests of model fit are well known.