Page 194, Chapter 5.2 Project 3

Part 1: Generate Uniformed Distributed Random INTEGERS


Write a computer program to generate uniformly distributed random integers in the interval m < x < n, where m and n are integers, according to the following algorithm:

Step 1 Let \(d=2^{31}\) and choose N (the number of random numbers to generate)
Step 2 Choose any seed integer \(Y\) such that \(10000 < Y < 999999\)
Step 3 Let \(i = 1\)
Step 4 Let \(Y = (15625Y + 22221) mod(d)\)
Step 5 Let \(X_i = m + floor[(n - m + 1)Y/d]\)
Step 6 Increment \(i\) by \(1: i = i + 1\)
Step 7 Go to Step 4 unless \(i = N + 1\)

Here, floor[p] means the largest integer not exceeding p.

Probability

As per definition, uniformly distributed random integers in the interval [m,n], will have an equal probability of being selected. If we let m = -500 and n = 500, then any number selected should have a probability of

\[P(X_i) = \frac{1}{n-m}\]

\[\frac{1}{500-(-500)} = \frac{1}{1000} = 0.001\]

Generate 5000 Uniformly Distributed Rand. Integers



Histogram


The histogram with 5000 integers shows a uniform random distribution with minimum value of -500 and maximum value of 500.

Density Plot


The density plot shows uniform probability of 0.001 as expected.

Part 2: Generate Uniformed Distributed Random NUMBERS



For most choices of , the numbers \(X_1\), \(X_2\),… form a sequence of (pseudo) random integers as desired. One possible recommended choice is Y = 568731. To generate random numbers (not just integers) in an interval a to b with a < b, use the preceding algorithm, replacing the formula step 5 by

Let \[X_i = a + \frac{Y(b-a)}{d-1}\]

Generate 5000 Uniformly Distributed Rand. Real Numbers (not only integers)



Histogram

Again, the histogram with 5000 real numbers shows a uniform random distribution with minimum value of -500 and maximum value of 500.

Density Plot


The density plot shows uniform probability of 0.001 as expected.

Page 233, Chapter 6.2 Project 1

Two alternative designs are submitted for a NASA Mars landing module. Which design would you recommend?
What assumptions are required? Are the assumptions reasonable?



Each system can be evaluated by separating it into components, calculating each component’s reliability, then calculating the total system reliability.

Part 1: Calculate the Reliability of System 1

\(A=0.993\)

\(B=R_1+R_2 - (R_1*R_2) = 0.995 + 0.995 -(0.995)^2 = 0.999975\)

\(C_1=L_{12}=L_1*L_2 = 0.99*0.999 = 0.98901\)

\(C=L_{12}+L_3 - (L_{12}*L_3)=L_{12}+0.998-(L_{12}*0.998)\)

\(C=0.98901+0.998-(0.98901*0.998)=0.99997802\)

\(D=0.98\)

\(E_1=0.99^2=0.9801\)

\(E_2=0.99^2=0.9801\)

\(E=E_1+E_2-(E_1*E_2)=0.9801+0.9801-0.96059601=0.99960399\)

Total System 1 Reliability = \(ABCDE=0.97270893\)

Part 2: Calculate the Reliability of System 2

\(A=0.998\)

\(B=R_1+R_2 - (R_1*R_2) = 0.995 + 0.995 -(0.995)^2 = 0.999975\)

\(C=R_s=R_1+R_2+R_3-R_1R_2-R_1R_3-R_2R_3+R_1R_2R_3\)

\(C=0.99+0.999+0.998-(0.99*0.999)-(0.99*0.998)-\)
\((0.999*0.998)+(0.99*0.999*0.998)=0.99999998\)

\(D=0.98\)

\(E=1-(1-B_1)(1-B_2)(1-B_3)(1-B_4)=1-0.01^4=0.99999999\)

Total System 2 Reliability = \(ABCDE= 0.9780155\)


The second system has the higher reliability and would be recommended.

Interactive Shiny App for Component Reliability


Test the app here.



Sources and Further Reading



Projects from A First Course in Mathematical Modeling by Giordano et al.

Practice Problems in Random Number Generation
Basics of Traditional Reliability
R Shiny Tutorial