Monte Carlo Theoretical

Question 1

Write the CDF for a Uniform distribution

Answer 1

Gerade mit Steigung U(r) = P( r <= R) = r

Question 2

Write the PDF for a uniform distribution

Answer 2

u(r) = d(U)/dr = 1

Question 3

Describe a mathematical procedure to generate seemingly
random numbers uniformly distributed between 0 and 1.

Answer 3

Use van Neumann algorithm where next random number is based on last random number.
x = (a*x(-1) + c) mod m

Question 4

Describe the computational steps of a Monte Carlo simulation for
the estimation of the number pi.

Answer 4

1. Compute random anlge theta
2. probability that needle intersects a line based on theta?

Test this out on paper

Question 5

Apply the inverse transform method to sample from a uniform
probability distribution U(a,b) defined in the range [a, b]

Answer 5

Look online for a random function, then calculate inverse

Question 6

Can you use the inverse transform method to sample failure
times from a probability function characterized by time-dependent
hazard rate? If so, provide a numerical example of such
probability function, show the analytical expression for the
inverse transform and comment it. What is the hazard rate?

Answer 6

Weilbull distribution has a time-dependant hazard-rate.
Write this down on paper.

alphaBeta^(t-1)e^(-Beta*t^(alpha))

hazard rate:
alpha * Beta * t^(alpha-1)

Question 7

Describe the steps to sample from a probability distribution fX(x)
defined in the interval [a,b] using the rejection method. What are
the conditions for the application of the rejection sampling
method?

Answer 7

We can use the rejection method by decomposing fx(x) into a simple pdf g(x) and a difficult to solve H(x). We need to be able determine the max of H(x).

1. Sample a x’ from the inverse of g(x) (or uniform)
2. put x’ into h(x’) = H(x)/maxH
3. Generate second random number R
4. if R <= h’(x) accept sample, otherwise reject

Question 8

Is it possible to sample from a probability distribution fX(x) defined in the
interval [a,b] that tends to infinity? If so, provide a numerical example of
such case and show how to sample from it using the rejection method.

Answer 8

It is possible to sample from a probability function that has a singularity.

Question 9

What is the definition of the efficiency of the rejection sampling method?
How can it be computed? What is a good strategy to increase the efficiency
of the rejection sampling method?

Answer 9

l

Question 10

What is the computational procedure to estimate the definite integral of a
function h(x) in the interval [a,b] using Monte Carlo? How can you show that
the Monte Carlo estimate is a good estimator for the true definite integral?

Question 11

Mathematically show that GN, i.e. the Monte Carlo estimate of the definite
integral of h(x)=g(x)*f(x) in the interval [a,b], is an unbiased and consistent
estimator for G, i.e. the true value of the definite integral of h(x) in the
interval [a,b].

Question 12

How can we increase the precision of GN , i.e. the Monte Carlo estimate of
the definite integral of h(x)=g(x)*f(x) in the interval [a,b]? What is the effect
of increasing N on the variance of the estimate Var[GN]? Derive an equation
that shows this effect.

Question 13

How can we increase the precision of GN , i.e. the Monte Carlo
estimate of the definite integral of h(x)=g(x)*f(x) in the interval
[a,b] using the same number of samples N? What is the effect of
the selection of g(x) on the variance of the estimate Var[GN]?
Derive an equation that shows this effect.

Question 14

What is the effect of selecting g(x) and f(x) functions that peak in
different regions of the [a,b] interval on the Monte Carlo estimate
of the definite integral of h(x)=g(x)*f(x) in the interval [a,b].

Question 15

What is the effect of biased sampling (Importance Sampling) on
the Monte Carlo estimate of the definite integral of h(x)=g(x)*f(x)
in the interval [a,b]. Show what importance sampling means in
practice with a graphical example.

Question 16

What is the implication of biased sampling (Importance Sampling)
from f1(x) on the award function g1(x) in the Monte Carlo estimate
of the definite integral of h(x)=g(x)*f(x) in the interval [a,b].

Question 17

What is a good selection of f1(x) in the biased sampling
(Importance Sampling) for the Monte Carlo estimate of the
definite integral of h(x)=g(x)*f(x) in the interval [a,b]. What is the
principle that should guide this selection? Motivate your answer
using a visual example and elaborate on the properties of the
resulting biased award g1(x).

Question 18

What is a good selection of f1(x) in the biased sampling
(Importance Sampling) for the Monte Carlo estimate of the
definite integral of h(x)=g(x)*f(x) in the interval [a,b]. What is the
principle that should guide this selection? Motivate your answer
using a visual example and elaborate on the properties of the
resulting biased award g1(x).

Question 19

What is the definition of the transition kernels T and C that
compose the transport kernel K describing the stochastic
transitions simulating the random walk of the system state in the
phase space.

Question 20

What do we sample from the kernels T and C that compose the
transport kernel K? How do we practically use T and C to evolve
the plant state as a random walk in the phase space

Question 21

How do we compute the reliability and point availability of a
system during the simulation of its life as a random walk in the
state space?

Question 22

How can we estimate the unreliability and the unavailability from
the knowledge of the transition density (t; k)

Question 23

Explain how we can reduce the solution of the transport equation
to a dart game and solve it via Monte Carlo simulation. What is
the probability function that we have to sample from?

Question 24

Discuss the difference between indirect and direct Monte Carlo
simulation. Which one requires fewer sampling calls to the
uniform distribution between [0,1]? Explain why.

Answer 24

In the indirect approach, we first sample transition time for all possible transitions and then sample which transition occurs.

In the direct approach we calculate all possible transitions directly and then order them on a time-scale. The first transition on the time scale occurs and we proceed to the next sample step. until we reach mission time.

The direct approach is better, when the individual components failure follows various distributions that are difficult to combine.

The indirect approach needs less sampling calls.

Question 25

Discuss advantages and disadvantages of direct and indirect
implementations of Monte Carlo simulations.

Answer 25

The advantage of direct sampling is, that it is more efficient. However

Question 26

Explain how we can use the spline method to numerically sample
from non-invertible probability functions. Discuss a high-level
code implementation for the spline sampling

Question 27

Explain in what sense the kernel K provides the local
knowledge about the system transport while the transition
density (t; k) provides the global knowledge about the system
transport.

Question 28

Given the system structure, the component transition rates
and the current state of the components, compute the
overall transport kernel out of the current system state for
the indirect implementation of the Monte Carlo method