exam

2. 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):

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

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

4. (2 points) 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.)

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

C2

0.85

C1

0.95

C3

0.90

C4

0.96

6-4. (2 points) 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?