Monte Carlo Experimental

Question 1

What are the steps for sampling random numbers from a continuous
probability function using the inverse transform method?

Answer 1

1. We create the inverse function of f(x) (a pdf)
2. Now we sample a random number from uniform distribution
3. We calculate the realization using the inverse function

Question 2

What are the conditions to apply the inverse transform method for
sampling continuous random numbers? Choose a continuous
probability function and show how to sample using this method

Answer 2

The probability density function has to be invertible.
Example: exponential distribution.

Question 3

Identify a continuous probability function for which the inverse transform
method for sampling cannot be applied and explain why it cannot be
applied.

Answer 3

A quadratic function cannot be inverted because it is not mapping values 1:1.

Question 4

Apply the inverse transform method to sample random numbers from
discrete probability distributions.

Answer 4

We simple invert the function stepwise. Do this on paper

Question 5

Given a network system whose functionality is defined by the
connectivity of two special nodes and the failure probabilities of each
component. Identify the minimal cut sets and detail the steps of a Monte
Carlo simulation to quantify the system reliability R.

Answer 5

We can either use a direct or indirect aproach.
In the direct approach we sample failures of the components and update the state after each sumple. once a state of the minimal cut set is reached, the system is failed until the end of the mission time. To assess reliability we run the monte carlo many times and create a reliability distribution.

Alternatively we can also follow an indirect approach where we can calculate a probability density function of transitions, sample the transition time and then the transition type.

Question 6

Describe the plant lifetime as a random walk in the state
space. What are the governing probabilities which drive the
stochastic path? What is their meaning?

Answer 6

the governing probabilites are the distribution T which indicates when a change in direction occurs and C which determines the direction of change.

Question 7

How do you quantify the reliability function R(t) and the availability functions A(t) of a system, if you can simulate its lifetime as a random walk in the state space?

Answer 7

For the reliability we count the number of trials that have no system failure before time t.

For the availability we seperate the function into dt increments and calculate what fraction of these dt increments have the system operational out of all increments.

Question 8

Given the reliability-block diagram of a system and the
failure rates of its components, detail the steps of a Monte
Carlo simulation to quantify the reliability function R(t) and
the availability functions A(t). Use the INDIRECT Monte
Carlo simulation and analog trials.

Question 9

Given the reliability-block diagram of a system and the
failure rates of its components, detail the steps of a Monte Carlo simulation to quantify the reliability function R(t) and the availability functions A(t). Use the DIRECT Monte Carlo
simulation and analog trials.

Question 10

Detail the steps to estimate the availability of a generic
system using the Monte Carlo simulation. How can we
quantify the availability A(t) at the specific time instant t?
Why do we call this value an estimate? How do we know
that this number is a good representative of the true
availability function?