Consider the set of 5 elements composed of two positive real numbers x and y, their arithmetic mean (x +y)/2, their geometric mean sqrt(xy), and their harmonic mean 2xy/(x + y).
set = {x, y, (x +y)/2, sqrt(xy), 2xy/(x + y) }
Now take the standard deviation of this set (sigma) and divide it by abs(x - y)
Q1: What is the limit of sigma/abs(x - y) taken as x-> y ?
(p.s. I know the answer already, It is a finite positive number that can be expressed with a simple analytic form (but not obvious). I am posting because I thought it was interesting)
Q2: Has anyone ever come across this question before, and if so, then where?
In the limit, the answer is obvious. The divisor is zero in the limit anyway.
No, I have never heard this question earlier.
The fact that the divisor goes to zero doesn't matter, the numerator also goes to zero. To evaluate the limit you actually have to work it out with calculus or numerically. The answer is not 1 or infinity.
Yes, yes. I overlooked this point. The variance itself tends to zero when x tends to y. Accordingly, it is of the zero by zero type of limit which is to be found using L'Hospital's rule.
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