Question

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.

Answer #1

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...

You are using a Monte Carlo simulation to estimate the area of
shape inside a unit square by taking n sample points. How big does
n have to be in order for the estimate to be within about ±0.01 of
the correct answer?
I believe this has to do with binomial distribution.

Perform Monte-Carlo estimation where theta = E(e^U) where U is a
continuous uniform random variable between zero and one a. Estimate
!using 1000 data points and obtain a 95% confidence interval for
this case. b. Perform part a) 100 times. Check how often the true
!actually does fall within the 100 resulting confidence intervals.
Please solve in R.

Conducting a Simulation
For example, say we want to simulate the probability of getting
“heads” exactly 4 times in 10 flips of a fair coin.
One way to generate a flip of the coin is to create a vector in
R with all of the possible outcomes and then randomly select one of
those outcomes. The sample function takes a vector of elements (in
this case heads or tails) and chooses a random sample of size
elements.
coin <- c("heads","tails")...

Phone calls can be classified as voice (V) if someone is
speaking, or data (D) if there is a modem or fax transmission.
Based on many observations made by the telephone company, we have
the following probability model: P [V] = 0.8, P [D] = 0.2. Data
calls and voice calls are produced independently. The random
variable Kn is the number of data calls in a collection of n calls
telephone
(a) What is the E [K100], the expected number...

This worksheet is about doing simulations on a TI-83/84, but
feel free to do the work on a computer if you prefer.
You are going to estimate the value of ? through a simulation. A
circle inscribed in a 1x1 square has area ?/4 (you may want to draw
a picture to convince yourself of this). Now, we can simulate
picking points inside the square as follows: randomly select a
value between 0 and 1 to be the x-coordinate, and...

In Mississippi, a
random sample of 110 first graders revealed that 48 of them have
been to a dentist at least once.
(a) Find a 95% confidence interval for the population proportion
of first-graders in Mississippi who have been to a dentist at least
once.
Enter the smaller number in the first box.
Confidence interval: ( , ).
College officials want
to estimate the percentage of students who carry a gun, knife, or
other such weapon. A 9898% confidence interval...

Problem 1: Relations among Useful Discrete Probability
Distributions. A Bernoulli experiment consists of
only one trial with two outcomes (success/failure) with probability
of success p. The Bernoulli distribution
is
P (X = k) =
pkq1-k,
k=0,1
The sum of n independent Bernoulli trials forms a binomial
experiment with parameters n and p. The binomial probability
distribution provides a simple, easy-to-compute approximation with
reasonable accuracy to hypergeometric distribution with parameters
N, M and n when n/N is less than or equal...

You plan to invest $4 million in the construction of an oil well
which has a potential yearly revenue of $10 million. The oil well
will be located in the Golf of Mexico. As we all know, this region
is constantly hit by hurricanes. Assuming that if during an entire
year there is a hurricane, this will disrupt your production and
your well will lose 20% its yearly production. And if during a year
there are two hurricanes, your well...

2) Airline accidents: According to the U.S. National
Transportation Safety Board, the number of airline accidents by
year from 1983 to 2006 were 23, 16, 21, 24, 34, 30, 28, 24, 26, 18,
23, 23, 36, 37, 49, 50, 51, 56, 46, 41, 54, 30, 40, and 31.
a. For the sample data, compute the mean and its standard error
(from the standard deviation), and the median.
b. Using R, compute bootstrap estimates of the mean, median and
25% trimmed...

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