Numerical Methods for Chemical Engineering: Applications in MATLAB
Written in a pedagogic style, this textbook unites the applications of numerical mathematics and scientific computing to the practice of chemical engineering.
TABLE OF CONTENTS Numerical Methods for Chemical Engineering: Applications in MATLAB
Confidence Intervals from the Approximate Posterior Density
The approximate posterior density ?( ?, ? y) for single-response data (8.116), describes the joint uncertainty in both ? and ?. Of more interest is the marginal posterior density for ?,
This is the posterior for ?, without regard to the exact value of ?. As we discard ? by integrating it out, it is called a nuisance parameter.
For the approximate posterior (8.116), the marginal posterior density can be calculated analytically,
Note again that for a linear model, this marginal posterior is exact.
Confidence interval for the mean of a population and the t-distribution
We now form confidence intervals for the model parameters and the predicted responses using this approximate marginal distribution. First, it is best to consider the simple model (8.104), y [ k ] = ? + ? [ k ]. After N measurements, the posterior density is
where the sample mean and sample variance are
The marginal posterior density for ? is
Defining the t-statistic,
whose distribution satisfies p( t ?) dt = p( ? y) d ?, we have
This is the famous t-distribution of Student, a pseudonym for W. S. Gosset ( Gosset, 1908). As the number of degrees of freedom ? approaches infinity, the t-distribution reduces to a
Gaussian (normal) distribution of mean zero and variance 1:
For finite...
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