2 For Spherical Uniform Distribution 21 2 points How can we

2. For Spherical Uniform Distribution

21. (2 points) How can we pick a set of random points uniformly distributed on the unit circle x₁2 + x₂2=1?

22. (2 points) How can we pick a set of random points uniformly distributed on the 4dimensional unit sphere

X₁^2+X₂²+X₃^2 +X₄^2+X₅^2=1?

Solution

1. first, we have to take two random numbers between -1 and 1. consider them as a pair.

let the pair is (x1,x2)

now, if x1² + x2² is less than or equal to 1, then the pair (x1, x2) is a random point from the unit circle x1²+x2²=1.

we can continue this process to get a set of random points from th unit circle.

2. first, we have to choose 5 points from -1 to 1. consider them as a 5-tuple.

let, such a 5-tuple is (x1, x2, x3, x4, x5).

now, if x12+x22+x32+x42+x52 is less than or equal to 1, we can consider the 5-tuple as a random point from the unit 5-dimensional sphere.

we can continue this procedure to get such points.