December 22 Information

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) reply

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) reply

2)There isn't enough information to answer Q2 (unless you make several unjustifiable assumptions). Dbfirs 08:17, 22 December 2012 (UTC) reply

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) reply

Given a rational function ${\displaystyle f}$, 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 ${\displaystyle f(i\omega )}$, where ${\displaystyle \omega }$ is the angular frequency of the source (for any ${\displaystyle \omega \in \mathbb {R} }$)? (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) reply

If functions f1 and f2 are impedances, then so are f1+f2 and (f1·f2)/(f1+f2). This defines the set of impedances recursively. Bo Jacoby ( talk) 19:44, 22 December 2012 (UTC). reply

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) reply

What is the impedance of such a circuit? Bo Jacoby ( talk) 08:12, 23 December 2012 (UTC). reply

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) reply