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.
![159_Image.png](https://secure.tutorsglobe.com/CMSImages/159_Image.png)
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