How do you choose how many points to sample in the circle? - This seems to be context-dependent...
Boomaa23
@rishiskhare I think you're right in that the number of points in the circle is context-dependent. I'd imagine that the greater the number of points, the more "accurate" your output function will be but at the expense of choosing more random points. In the example of using this to calculate pi, your estimate of pi will become more accurate as the number of points increases.
RishSharma7
Can somebody maybe give me a better explanation for this "wrong method?" Didn't quite understand why it was named so in lecture. Also, Rishi's question is one that I also had. At what point do we generally say that we have "enough" points for sampling in the circle, and what factors tend to play into that decision?
buggy213
it is wrong because it doesn't uniformly sample points in the circle, it tends to sample points near the center of the circle. the illustration on the next slide is a good example why. as a concrete example, if you consider $$P[0<r<0.1]$$, we uniformly sample $$r$$, so we have a 10% chance of sampling a point in this range. However, this region is only 1% of the total area of the circle. so, our sampled points will be denser near the center.
How do you choose how many points to sample in the circle? - This seems to be context-dependent...
@rishiskhare I think you're right in that the number of points in the circle is context-dependent. I'd imagine that the greater the number of points, the more "accurate" your output function will be but at the expense of choosing more random points. In the example of using this to calculate pi, your estimate of pi will become more accurate as the number of points increases.
Can somebody maybe give me a better explanation for this "wrong method?" Didn't quite understand why it was named so in lecture. Also, Rishi's question is one that I also had. At what point do we generally say that we have "enough" points for sampling in the circle, and what factors tend to play into that decision?
it is wrong because it doesn't uniformly sample points in the circle, it tends to sample points near the center of the circle. the illustration on the next slide is a good example why. as a concrete example, if you consider $$P[0<r<0.1]$$, we uniformly sample $$r$$, so we have a 10% chance of sampling a point in this range. However, this region is only 1% of the total area of the circle. so, our sampled points will be denser near the center.