Problem 1 - Spherical Uniform Distribution (Hint: Knuth's idea introduced in Section 4 of the attached article written by Jan Poland. You do not have to explain why):

Article- Three Different Algorithms for Generating Uniformly Distributed Random Points on the N-Sphere by Jan Poland

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

2. How can we pick a set of random points uniformly distributed on the 4-dimensional unit sphere x₁² + x₂²+ x₃²+ x₄²+ x₅²=1?

Problem 2 - Show the mean and the variance of the triangular distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. (You must show why.)

Problem 3- Consider the following system made up of functional components in parallel and series.

Compute and compare the probabilities that the system fails when the probability that component C1 functions is improved to a value of 0.98 and when the probability that component C2 functions is improved to a value of 0.88. Which improvement increases the system reliability more?

Attachment:- Article.rar