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#1 2013-12-06 01:56:52

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Finding average distance (circle)

From http://www.mathisfunforum.com/viewtopic … =19518&p=5 #117

A point 'P' inside a unit circle is chosen at random.
a) A direction is chosen at random at P and a line is drawn from P to the point in the circumference 'Q' in that direction. What is the expected length of the line segment PQ ?
b)  A direction is chosen at random at the center and a line is drawn from P to the point in the circumference 'Q' in that direction. What is the expected length of the line segment PQ ?

Analytical answer is given in #147.

Here, we'll see how to do a simulation in J.
Suppose 'r' is the distance between P and center of the circle, and θ is the angle chosen,
we can write the line segment length as


and

for (a) and (b), respectively.
r^2 must be uniformly distributed in [0,1], since area is proportional to r^2.

load 'trig'
r=:%:?10000000#0 
th=:2p1*?10000000#0
(+/%#)(%:(1-(r*sin th)^2))-r*cos th  NB. appx. 0.848544
(+/%#) %:(((r-cos th)^2)+(sin th)^2) NB. appx. 1.131613

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#2 2013-12-06 02:44:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,894

Re: Finding average distance (circle)

Hi gAr;

Could you pseudocode that?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#3 2013-12-06 03:01:21

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Finding average distance (circle)

Hi,

Perhaps the R code would be more readable:

r=sqrt(runif(10000000,0,1))
th=2*pi*runif(10000000,0,1)
mean(sqrt(1-(r*sin(th))^2)-r*cos(th))   # 0.8485681
mean(sqrt(((r-cos(th))^2)+(sin(th))^2)) # 1.131671

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#4 2013-12-06 05:45:34

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,894

Re: Finding average distance (circle)

Hi gAr;

Okay got it.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

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