It’s not just for computational convenience.

rms SD minimizes expected distance from mean while avg of abs val does it for the median

It’s not just for computational convenience.

rms SD minimizes expected distance from mean while avg of abs val does it for the median

This includes some useful insight into how to read and interpret modal logic.

Source: *(1000) John Gould’s answer to What do atheists think of Gödel’s proof that God exists? – Quora*

The question is ill-posed because there are many different ways of choosing two spots “at random”. BUT if we assume that the cuts are made independently with each chosen according to a uniform probability per unit length then the answer is the fraction of the big square that is not shaded in the diagram below . (ie p = 1-(17/20)^2)

Since this is tagged with “Puzzles and Trick Questions” it may be that I am missing something. But my answer would be [math]\Sigma_{m=0}^{n-1}p^{2n-1-m}(1-p)^m[/math] .

This follows the pattern of the best 2 out of 3 case where Alice has to win either two or three games – which happens in cases lww,wlw,wwl or www with probability [math]3p^2(1–p)+p^3=3p^2–2p^3[/math] (where the fact that the game may be stopped when she wins twice just corresponds to the fact that [math]pp(1-p)+ppp=p^2[/math] , and the same answer is obtained by taking the complement of the cases where Bob wins either 2 or 3 games).

This question has been around for a while and has some decent answers. But I want to suggest a simpler and more intuitive version. (And it will be easier to follow if I replace the variable [math]x[/math] by [math]t[/math] so as not to confuse it with the real part of the complex function value.)

First, to define [math]f(z)=e^z[/math] without using trigonometry or ever mentioning trig functions, we can use either the power series or the complex differential equation [math]f’=f[/math] with [math]f(0)=1[/math]. And either way we get [math]\frac{d}{dt}e^{it}=ie^{it}[/math].

Now multiplication by [math]i[/math] just rotates the complex plane by a right angle, so the curve in the plane given parametrically by [math](x(t),y(t))[/math] with [math]x(t)+iy(t)=e^{it}[/math] has a tangential velocity vector which is always perpendicular to its position vector and equal in magnitude.

Since it starts at [math]t=0[/math] at [math](x,y)=(1,0)[/math] the curve is just the unit circle centred at the origin.

And since its velocity vector is always of length 1, if we think of the parameter [math]t[/math] as representing time, then the point moves with speed 1 and so the time taken to complete a circuit, ie the period of [math]e^{it}[/math], is just the same as the circumference of the unit circle (commonly denoted by [math]2\pi[/math] ).

The question of whether, and if so how, our brains are *Wired for Numbers* has obvious implications for the teaching of mathematics, and also, I think, for the question of how we interpret the “reality” of number and quantity in a “philosophy of math” sense.

Source: *Is Your Brain Wired for Numbers? | The Scientist Magazine®*

Welcome to Alan’s Math Notes. This is where I am planning to restore and make available various on-line notes and learning resources that I either developed myself and/or found useful and freely available from other sources.