Fundamentals of Probability and Statistics for Engineers

moments of X can be directly generated from Equation (9.77). We remark that, although an estimator for^2 is not required, it is ...

produces a moment estimator for in the form Although this estimator may be inferior to 1/X in terms of the quality criteria we h ...

where is the Lagrange multiplier. Taking the first variation of Equation (9.88) and setting it to zero we obtain as a condition ...

Three moment estimators for ,and (3)– can be found by means of establishing and solving the first three moment equations. Let 2 ...

Let f(x; ) be the density function of population X where, for simplicity, is the only parameter to be estimated from a set of sa ...

To see that this procedure is plausible, we observe that the quantity is the probability that sample X 1 ,X 2 ,...,Xn takes valu ...

Analogous results are obtained when population X is discrete. Furthermore, the distribution of tends to a normal distribution as ...

Solving the above equations simultaneously, the MLEs of m and^2 are found to be and The maximum likelihood estimators for m and^ ...

and the maximum likelihood estimator for is This estimator is seen to be different from that obtained by using the moment method ...

The mean and variance of are We see that is biased but consistent. Ex ample 9. 17. Let us now determine the MLE of r^2 in Exampl ...

where I 1 is the modified first-order Bessel function of the first kind, and As we can see, although likelihood equations can be ...

The standardized random variable U, defined by is then N (0, 1) and it has pdf Suppose we specify that the probability of U bein ...

We now make several remarks concerning the foregoing definition. R emark 1: as we see from Equation (9.126), confidence limits a ...

Hence, using the transformation given by Equation (9.123), we have the general result This result can also be used to estimate m ...

betweenX and m can be at most equal to one-half of the interval width. We thus have the result given in Theorem 9.6. Theorem 9.6 ...

If U is N(0, 1), V is^2 -distributed with n degrees of fr eedom, and U and V are independent, then the pdf of T has the form Thi ...

Returning to random variable Y defined by Equation (9.132), let and Then where U is clearly distributed according to N(0,1). We ...

Example 9.19.Problem: let us assume that the annual snowfall in the Buffalo area is normally distributed. Using the snowfall rec ...

9.3.2.3 Confidence Interval for^2 in N(m^2 ) An unbiased point estimator for population variance^2 is S^2. For the con- structio ...

number of degrees of freedom can be determined by interpolation from tabu- lated values of the PDF of the chi-squared distributi ...