[SOLVED] Probability & statistics for eecs homework 08

1. Use the method of inverse transform sampling (or called the method of inverse CDF)
to obtain samples from each of the following continuous distributions:
(a) Logistic distribution with CDF F(x) = 1
1+e−x , ∀x ∈ R.
(b) Rayleigh distribution with CDF F(x) = 1 − e−x2/2, ∀x > 0.
(c) Exponential distribution with CDF F(x) = 1 − e−x, x > 0.2. Develop algorithms to obtain samples from each of the following discrete distributions:
(a) Bernoulli distribution Bern(0.5).
(b) Binomial distribution Bin(20, 0.5).
(c) Geometric distribution Geom(0.5).
(d) Negative Binomial distribution NBin(10, 0.5).3. Let Ui ∼ Unif(0, 1), i ≥ 1 be i.i.d. random variables. Define N as follows:
N = max
�
n :
�n
i=1
Ui ≥ e−1
·
�
(a) Estimate E(N) by generating 5000 values of N and then use the sample mean.
(b) Estimate Var(N).
(c) Estimate P(N = i), for i = 0, 1, 2, 3.
(d) Can you find the exact distribution of N4. (a) In the classical three-door Monty Hall problem, a contestant chooses one of three
closed doors, two of which have a goat behind them and one of which has a car.
Monty, who knows where the car is, then opens one of the two remaining doors.The door he opens always has a goat behind it (he never reveals the car). If he
has a choice, then he picks a door at random with equal probabilities. Monty
then offers the contestant the option of switching to the other unopened door.
If the contestants goal is to get the car, should she switch doors? Please use
simulation to compare the strategy of never-switching and the strategy
of switching.(b) Now we consider the progressive Monty Hall problem. This time we assume there
are n identical doors, where n is an integer satisfying n ≥ 3. One door conceals a
car, the other n − 1 doors conceal goats.You choose one of the doors at random
but do not open it. Monty then opens a door he knows to conceal a goat, always
choosing randomly among the available doors. At this point he gives you the option
either of sticking with your original door or switching to one of the remaining doors.You make your decision. Monty now eliminates another goat-concealing door (at
random) and once more gives you the choice either of sticking or switching. This
process continues until only two doors remain in play.What strategy should you
follow to maximize your chances of winning? We consider three strategies:
i. Select a door at random and stick with it throughout.
ii. Select a door at random, then switch doors at every opportunity, choosing
your door randomly at each step.
iii. Select a door at random, stick with your first choice until only two doors
remain, and then switch.
When n = 4 and n = 100, please use simulation to compare such three
strategies.5. Estimate the value of the percolation threshold via Monte Carlo simulation.
(a) Percolation. Given a composite systems comprised of randomly distributed insulating and metallic materials: what fraction of the materials need to be metallic
so that the composite system is an electrical conductor? Given a porous landscape
with water on the surface (or oil below), under what conditions will the water be
able to drain through to the bottom (or the oil to gush through to the surface)?
Scientists have defined an abstract process known as percolation to model such
situations.(b) The model. We model a percolation system using an n-by-n grid of sites. Each
site is either open or blocked, where open means the water or other materials can
flow through such site. Open sites can be further classified into two categories:
full open sites and empty open sites. A full open site is an open site that can be
connected to an open site in the top row via a chain of neighboring (left, right,
up, down) open sites.An empty open site is an open site that is NOT full. We
say the system percolates if there is a full open site in the bottom row. In other
words, a system percolates if we could find a path of open sites from the top row
to the bottom row. (For the insulating/metallic materials example, the open sites
correspond to metallic materials, so that a system that percolates has a metallic
path from top to bottom, with full sites conducting. For the porous substance
example, the open sites correspond to empty space through which water might
flow, so that a system that percolates lets water fill open sites, flowing from top to
bottom.)Figure 1: The blocked sites are displayed in black. The full open sites are displayed in cyan.
The empty open sites are displayed in white.
(c) The problem. In a famous scientific problem, researchers are interested in the
following question: given an n-by-n random grid, if sites are independently set to
be open with probability p (and therefore blocked with probability 1 − p), what
is the probability that the system percolates? When p = 0, the system does not
percolate; when p = 1, the system percolates. The plots below show the site
vacancy probability p versus the percolation probability for 20-by-20 random grid
(left) and 100-by-100 random grid (right).When n is sufficiently large, there is a threshold value p∗ such that when p < p∗
a random n-by-n grid almost never percolates, and when p > p∗, a random n-byn grid almost always percolates. No mathematical solution for determining the
percolation threshold p∗ has yet been derived. Your task is to write a computer
program to estimate p∗.(d) Monte Carlo simulation. To estimate the percolation threshold, consider the
following computational experiment:• Initialize all sites to be blocked and black.
• Repeat the following until the system percolates:
– Choose a site uniformly at random among all blocked sites.
– Open the site.
• The fraction of sites that are opened when the system percolates provides an
estimate of the percolation threshold.For example, if sites are opened in a 20-by-20 lattice according to the snapshots
below, then our estimate of the percolation threshold is 204/400 = 0.51 because
the system percolates when the 204th site is opened.By repeating this computation experiment T times and averaging the results, we
obtain a more accurate estimate of the percolation threshold. Let xt be the fraction
of open sites in computational experiment t. The sample mean x¯ provides an
estimate of the percolation threshold:
x¯ = x1 + x2 + · · · + xT
T .(e) The task. Estimate the percolation threshold for 20-by-20 grid, 50-by-50 grid,
and 100-by-100 grid via Monte Carlo simulation.