Wikipedia:Reference desk/Archives/Mathematics/2012 December 22

= December 22 =

Stats
1) If several tests are administered, and the class average is always below the passing grade, does that mean one person will fail?

2) If you were given a 73% average on one test and a 78% average on another, and noone got above 95% on either test, what is the probability that one person failed both tests?Curb Chain (talk) 03:38, 22 December 2012 (UTC)


 * I numbered your Q's for ease of responders:


 * 1) I'd expect far more than 1 to fail, except that the class average being below the passing grade means either something is seriously wrong or they have yet to apply a grading curve. StuRat (talk) 06:32, 22 December 2012 (UTC)


 * 2)There isn't enough information to answer Q2 (unless you make several unjustifiable assumptions).  D b f i r s   08:17, 22 December 2012 (UTC)

The answer to the first question is yes: at least one person will fail. For the class average to be below the passing grade p, the sum of n students' scores on the test has to be ≤ np–1; over k tests the cumulative score of all students combined has to be ≤ knp–k. On the other hand, for each student to get a passing average, each student has to get a cumulative score on all k tests combined of at least pk, so the cumulative score of all students combined has to be ≥ pkn. Since these inequalities contradict each other, someone has to fail. Duoduoduo (talk) 21:05, 22 December 2012 (UTC)

Possible impedances as function of frequency
Given a rational function $$f$$, is there a way to determine whether there is an electrical circuit built of resistors, capacitors, and inductors (ideal and with positive resistance/capacitance/inductance) which its impedance is $$f(i\omega)$$, where $$\omega$$ is the angular frequency of the source (for any $$\omega \in \mathbb{R}$$)? (I know this is sounds physical, but I think the problem is to determine which functions are possible as impedance and which functions are not, and this is more "mathematical", as it requires some proof) --77.125.85.139 (talk) 19:21, 22 December 2012 (UTC)
 * If functions f1 and f2 are impedances, then so are f1+f2 and (f1&middot;f2)/(f1+f2). This defines the set of impedances recursively. Bo Jacoby (talk) 19:44, 22 December 2012 (UTC).
 * But why every impedance function can be obtained in such a way? Remember that there are circuits which are neither series, nor parallel. --77.125.85.139 (talk) 20:12, 22 December 2012 (UTC)
 * What is the impedance of such a circuit? Bo Jacoby (talk) 08:12, 23 December 2012 (UTC).


 * Possibly a better question for WP:Reference desk/Science. My gut says the realizability with components with the given constraints will be determined by the poles and zeros of the function throughout the complex plane. There must be a finite number of them, counting multiplicity (but this is implied by the function being rational). The non-real poles must all occur in conjugate pairs, which I think is the same as saying that all the coefficients of the rational function must be real. The function must be non-negative at zero. And the real part of every pole must probably be non-negative. Or something like that. — Quondum 07:08, 24 December 2012 (UTC)