MATH 407: The Project
Problem 1. Find a procedure for sampling uniformly on the surface of the sphere.
(a) Use computer to generate a thousand points that are random, independent, and uniform on the unit sphere, and print the resulting picture.
(b) By putting sufficiently many independent uniform points on the surface of the Earth (not literally but using a computer model, of course), estimate the areas of Antarctica and Africa, compare your results with the actual values, and make a few comments (e.g. are the relative errors similar? would you expect them to be similar? if not, which one should be bigger? etc.)
Problem 2. Get a computer program for distinguishing a randomly generated sequence of zeroes and ones from a cooked-up one. You are welcome to write the program yourself or use a ready one you find on the web or in some book. Test your program on the following two sequences: the sequence consisting of the concatenation of all numbers in binary form1
0110111001011101111000…
and a similar sequence consisting of the concatenation of all prime numbers in binary form
0110111011111011 …
The first sequence (the fractional part of the Champernowne number) is known to be random when considered in base 10; the second sequence (the fractional part of the Copeland-Erd ̈os constant in binary form) is known to be random. In both cases, randomness is understood in a very specific way, and you are welcome to discuss this point too.
1spacing is introduced only for convenience: to indicate how the numbers are appearing 1