By A.M. Mathai
Textbook for a one-semester graduate path for college students focusing on mathematical statistics or in multivariate research, or reference for theoretical in addition to utilized statisticians, confines its dialogue to quadratic varieties and moment measure polynomials in genuine general random vectors and matr
Read Online or Download Quadratic Forms in Random Variables: Theory and Applications (Statistics: a Series of Textbooks and Monographs) PDF
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Extra resources for Quadratic Forms in Random Variables: Theory and Applications (Statistics: a Series of Textbooks and Monographs)
For example, the polynomial equation x 3 Ϫ 3x 2 ϩ 2x ϭ 0 may be rewritten in the form x(x2 Ϫ 3x ϩ 2) ϭ 0 or x(x Ϫ 1)(x Ϫ 2) ϭ 0 Since the product of two real numbers can be equal to zero if and only if one (or both) of the factors is equal to zero, we have xϭ0 xϪ1ϭ0 or xϪ2ϭ0 from which we see that the desired roots are x ϭ 0, 1, and 2. The Quadratic Formula In general, the problem of finding the roots of a polynomial equation is a difficult one. But the roots of a quadratic equation (a polynomial equation of degree 2) are easily found either by factoring or by using the following quadratic formula.
Thus, 0 x 0 Յ 5 means Ϫ5 Յ x Յ 5 (Figure 5a). To obtain an alternative solution, observe that 0 x 0 is the distance from the point x to zero, so the inequality 0 x 0 Յ 5 implies immediately that Ϫ5 Յ x Յ 5. Solution x FIGURE 5 –5 (a) x 5 0 –5 (b) 0 5 Next, the inequality 0 x 0 Ն 5 states that the distance from x to zero is greater than or equal to 5. This observation yields the result x Ն 5 or x Յ Ϫ5 (Figure 5b). EXAMPLE 14 Solve the inequality 0 2x Ϫ 3 0 Յ 1. Solution The inequality 0 2x Ϫ 3 0 Յ 1 is equivalent to the inequalities Ϫ1 Յ 2x Ϫ 3 Յ 1 (see Example 13).
MEETING PROFIT GOALS A manufacturer of a certain commodity has estimated that her profit in thousands of dollars is given by the expression Ϫ6x2 ϩ 30x Ϫ 10 where x (in thousands) is the number of units produced. What production range will enable the manufacturer to realize a profit of at least $14,000 on the commodity? 90. 08 mg/cc. 91. COST OF REMOVING TOXIC POLLUTANTS A city’s main well was recently found to be contaminated with trichloroethylene (a cancer-causing chemical) as a result of an abandoned chemical dump that leached chemicals into the water.