Yeah thats the way I did it with the angle as the parameter, which gives 4/pi. I still find it weird though that it yields a different answer with distances, using:4/pi sounds about right. Been a while so I can't remember.
But the average definition used is:
\,d\theta)
Oh right, so it has to do with the different probability distributions of the chords. But yeah I would imagine (not 100% sure) the average of any chord would be the same as the average of all horizontal chords, because you can do the exact same infinite set of chords at any inclanation from the horizontal on a circle, which is why just doing it with the angles gives the right answer of 4pi/3. It goes quite deep lol.
h is not uniformally distributed. You need to take into account the probability density function of h.Yeah thats the way I did it with the angle as the parameter, which gives 4/pi. I still find it weird though that it yields a different answer with distances, using:
Oh well, the angle way is more intuitive anyway lol.
Lol I don't know anything about uni probability but I'll take your word for ith is not uniformally distributed. You need to take into account the probability density function if h.
Ahhhhhhh that makes so much more sense. I was thinking it may have something to do with that uneven distribution of chord lengths with perp distance, but didn't know how to put it into words or even if that was a thing. Thanks for clearing up that misconceptionIt's essentially just saying thatis uniformly distributed, resulting in in a uniform distribution of lengths.
Consider this example: Suppose you wanted to plot an equal number of points around a unit circle. If you calculated the (x,y) coordinates from equal increments of theta, you'll end up with a uniform distribution of points. Now, try doing it by generating the y coordinates from equal increments of x coordinates. You'll find that the points aren't uniformly distributed - if you were to imagine plotting a huge number of points around the circle, most of the points will be skewed away from the left and right ends, because a small change in x there will lead to a large change in y (since the circle is very steep at x = +-1). In this case, the points are not uniformly distributed because the probability of picking a point with a small y value (near zero) will be much smaller than picking a point with a large y value (near 1) since most of the points have y-values skewed towards 1.
In our case, parameterisation by theta is correct since theta results in a uniform distribution. If you try to parameterise by h, you're more likely to get a longer chord length than a shorter chord length due to the steepness of the circle when h is small (this is analogous to the example above if you try to parameterise using x). When h is small, small changes to h result in small changes in L. When h is large, small changes in h result in large changes in L. Thus, the chord lengths are more skewed towards the larger values since there's a lot more closely spaced L values when h is small. This is also evident from the calculation: parameterisation by theta results in a value of 4/pi = 1.27, whereas parameterisation by h results in a value of pi/2 = 1.57. The value is significantly larger due to this skewed distribution.
Is the answer
Yes.Is the answer?
Sweet. I basically solved it using sine rule on the different triangles:
What course you planning on doing (if you want to say)?
I sure hope that geometry doesn't have more of a weight in this exam, convoluted geo questions are the absolute worst.
YesIs the answer to that problemthough?
Wait the one just posted or the one before that which I said was