Fundamentals of Probability and Statistics for Engineers

and (11.23) with the Cramr–Rao lower bounds defined in Section 9.2.2. In order to evaluate these lower bounds, a probability dis ...

This completes the proof. The theorem stated above is a special case of the Gauss–Markov theorem. Another interesting comparison ...

H ence,^2 is unbiased with k 1/(n 2), giving or, in view of Equation (11.30), Example 11.2.Problem: use the results given in Exa ...

Answer: in this case, we have n 14. The quantities of interest are The substitution of these values into Equations (11.7), (11.8 ...

.R esult i: let^2 be the unbiased estimator for^2 as defined by Equation (11.33), and let It follows from the results given in S ...

and where, as seen from Equations (11.20), (11.22), and (11.23), and are, (^2) estimated by (^2). The derivation given in Sectio ...

.R esult 2: a [100(1 )% confidence interval for is determined by [see Equation (9.141)] .Result 3: a [100(1 )]% confidence inter ...

Answer: equation (11.41) gives the desired confidence limits, with n 14, The observed confidence limits are thus given by This r ...

Using as the test statistic, we have shown in Section 11.1.4 that the random variable defined by Equation (11.36) has a t- distr ...

Similarly, significance tests about the value of can be easily carried out with use of as the test statistic. An important speci ...

Since 0. That is, we conclude that the data do not indicate a linear relationship between E Y and x; the probability that we are ...

If we let and Equation (11.47) can be represented by vector–matrix equation: Comparing Equation (11.48) with Equation (11.12) in ...

Answer: in this case, C is a 12 3 matrix and We thus have, upon finding the inverse of CTC by using either matrix inversion form ...

Confidence intervals for the regression parameters in this case can also be established following similar procedures employed in ...

Answer: let x 1 x, x 2 x^2 ,andlet The least-square estimate for is given by Equation (11.49), with and Thus or Table 11. 5 Popu ...

Let us note in this example that, since x 2 x^21 , matrix C is constrained in that its elements in the third column are the squa ...

where wi are assigned weights. In vector–matrix notation, show that estimates and now take the form where 11.5 (a) In simple lin ...

determine the estimated regression line for Y as a function of log 10 v. 11.8 An experimental study of nasal deposition of parti ...

(b) Estimate 11.11 In Problem 11.7, when vehicle weight is taken into account, we have the multiple linear regression equation w ...

Some available data are presented in Table 11.13. Determine the least-square estimate of the regression parameters. Table 11.13 ...