In: Math

A Monte Carlo simulation is a method for finding a value that is difficult to compute by performing many random experiments.

For example, suppose we wanted to estimate π to within a certain accuracy. We could do so by randomly (and independently) sampling n points from the unit square and counting how many of them are inside the unit circle (assuming that the probability of selecting a point in a given region is proportional to the area of the region). By assuming we actually get the expected number, we can solve for π.

(a) Describe a reasonable sample space to model this experiment.

(b) Let N be the number of sample points that are inside the unit circle. Find E(N).

(c) Use this to construct a random variable P with E(P) = π. This random variable will give your estimate of π.

(d) Find the variance of P.

(e) Use Chebychev’s inequality to find a value of n that guarantees your estimate is within 1/1000 of π with probability at least 50%.

Ans) a) A reasonable sample space to model this experiment is the unit square i.e, -

b) Let N be the number of sample points that are inside the unit circle. Given that the probability of selecting a point in a given region is proportional to the area of the region. Therefore probability of selecting a point in the unit circle is as area of the unit circle is and the area of the unit square is 1.

Let there are n sample points.Therefore if a sample point falls inside the unit circle then we have a success, otherwise we have a failure. So,

c)

d)

e) By Chebyshev's inequality

Given

Let as we put the estimate of as P .

Alternatively,

Given So,

Now here we are assuming the distribution of P to be normal as n is large

As

where is the upper th point of N(0,1)

Here is unknown so we estimate it by

otherwise

Describe the steps needed to perform the Monte Carlo
Simulation.

Make a program in C++ and write explanations. Monte-Carlo
methods Calculating pi with Monte-Carlo method is NOT allowed.

This problem is also a Monte Carlo simulation, but this time in
the continuous domain: must use the following fact: a circle
inscribed in a unit square
has as radius of 0.5 and an area of ?∗(0.52)=?4.π∗(0.52)=π4.
Therefore, if you generate num_trials random points in the unit
square, and count how many land inside the circle, you can
calculate an approximation of ?
For this problem, you must create code in python
(A) Draw the diagram of the unit square...

preform a Monte Carlo simulation in R to generate the
probability distribution of the sum of two die (for example 1st die
is 2 and second die is 3 the random variable is 2+3=5). The
R-script should print out (display in R-studio) or have saved files
for the following well labeled results:
1. Histrogram or barchart of probability distribution
2. Mean of probability distribution
3. Standard deviation of probability distribution

This problem is also a Monte Carlo simulation, but this time in
the continuous domain: must use the following fact: a circle
inscribed in a unit square
has as radius of 0.5 and an area of ?∗(0.52)=?4.π∗(0.52)=π4.
Therefore, if you generate num_trials random points in the unit
square, and count how many land inside the circle, you can
calculate an approximation of ?
For this problem, you must create code in python
(B) Without drawing the diagram, calculate the value...

Please provide an example of a Monte Carlo simulation model (and
explain).
Your simulation example should be able to:
Tackle a wide variety of problems using simulation
Understand the seven steps of conducting a simulation
Explain the advantages and disadvantages of simulation
Develop random number intervals and use them to generate
outcomes
Understand alternative computer simulation packages
available
Explain the different type of simulations
Explain the basic concept of simulation

1. Compare paramatric, historical, and monte carlo simulation
methods in identifying VaR (value at risk)
2. What are the pros and cons of those?
3. Identify some weights on historical losses that you think
should make more sense in current trading war environment.

Discuss about Monte Carlo simulation, with a focus on its
steps.
(100% rating)

problem is also a Monte Carlo simulation, but this time in the
continuous domain: must use the following fact: a circle inscribed
in a unit square
has as radius of 0.5 and an area of ?∗(0.52)=?4.π∗(0.52)=π4.
Therefore, if you generate num_trials random points in the unit
square, and count how many land inside the circle, you can
calculate an approximation of ?
For this problem, you must create code in python
(A) Draw the diagram of the unit square with...

How
to do the monte carlo method on a calculator? step by step
method.

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