Recall the rv X= amount of gravel (in tons) sold by a...

Recall the rv $X=$ amount of gravel (in tons) sold by a construction supply company in a given week from Example $3.9,$ whose pdf is $$f(x)=\left\{\begin{array}{cc}{\frac{3}{2}\left(1-x^{2}\right)} & {0 \leq x \leq 1} \\ {0} & {\text { otherwise }}\end{array}\right.$$ Consider simulating values from this distribution using the accept-reject method with a Unif[0, 1] "candidate" distribution, i.e., $g(x)=1$ for $0 \leq x \leq 1 .$ (a) Find the smallest majorization constant $c$ so that $f(x) / g(x) \leq c$ for all $x$ in $[0,1]$ . (b) Write a program to simulate values from this distribution. (c) On the average, how many candidate values must your program generate in order to create $10,000$ "accepted" values? (d) Simulate $10,000$ values from this distribution, and use to estimate the mean $\mu$ of this distribution. How close is your sample mean to the true value of $\mu$ (which you can determine using the appropriate integral)? The supply company's management looks at quarterly data for $X,$ i.e., values $X_{1}, \ldots, X_{13}$ for 13 weeks (one quarter of a year). Of particular interest is the variable $M=\min \left(X_{1}, \ldots, X_{13}\right)$ the least amount of gravel sold in one week during a quarter. Use your program in (b) to simulate the rv $M,$ and use the results of at least $10,000$ simulated values of $M$ to estimate $P(M<.1),$ the chance that the worst sales week in a quarter saw less than .1 tons of gravel sold. [Hint: Simulate each $X_{i} 10,000$ times for $i=1, \ldots, 13,$ and then compute the minimum of each set of 13 values to create a value for $M . ]$

Recall the rv $X=$ amount of gravel (in tons) sold by a construction supply company in a given week from Example $3.9,$ whose pdf is
$$f(x)=\left\{\begin{array}{cc}{\frac{3}{2}\left(1-x^{2}\right)} & {0 \leq x \leq 1} \\ {0} & {\text { otherwise }}\end{array}\right.$$
Consider simulating values from this distribution using the accept-reject method with a Unif[0, 1] "candidate" distribution, i.e., $g(x)=1$ for $0 \leq x \leq 1 .$
(a) Find the smallest majorization constant $c$ so that $f(x) / g(x) \leq c$ for all $x$ in $[0,1]$ .
(b) Write a program to simulate values from this distribution.
(c) On the average, how many candidate values must your program generate in order to create
$10,000$ "accepted" values?
(d) Simulate $10,000$ values from this distribution, and use to estimate the mean $\mu$ of this distribution. How close is your sample mean to the true value of $\mu$ (which you can determine using the appropriate integral)?
The supply company's management looks at quarterly data for $X,$ i.e., values $X_{1}, \ldots, X_{13}$ for 13 weeks (one quarter of a year). Of particular interest is the variable $M=\min \left(X_{1}, \ldots, X_{13}\right)$ the least amount of gravel sold in one week during a quarter. Use your program in (b) to simulate the rv $M,$ and use the results of at least $10,000$ simulated values of $M$ to estimate $P(M<.1),$ the chance that the worst sales week in a quarter saw less than .1 tons of gravel sold. [Hint: Simulate each $X_{i} 10,000$ times for $i=1, \ldots, 13,$ and then compute the minimum of each set of 13 values to create a value for $M . ]$

Key Concepts

Accept-Reject Method

The accept?reject method is a Monte Carlo simulation technique used to generate observations from a target distribution by sampling from a candidate (proposal) distribution that is easier to sample from. A candidate value is proposed and then accepted with a probability that ensures the samples follow the target density, ensuring proper weighting relative to the candidate density.

Candidate Distribution

In the context of simulation, a candidate distribution is an easily sampled distribution used as an auxiliary step to generate values from the desired (target) distribution. Its density function should be comparable to the target density, and typically a uniform distribution or another simple distribution is chosen for ease of implementation.

Majorization Constant

The majorization constant is a constant c chosen so that the target density is always less than or equal to c times the candidate density over the support. It ensures that for every candidate sample, the probability of acceptance (the ratio of the target and scaled candidate densities) does not exceed one, making the acceptance step valid in the accept?reject method.

Monte Carlo Simulation

Monte Carlo simulation involves generating random samples to approximate numerical quantities, such as integrals, expected values, or probabilities. It is particularly useful when analytical solutions are difficult to obtain, and it relies on the law of large numbers to provide accurate approximations as the number of simulations increases.

Estimation of Expected Value

Estimating the expected value (or mean) of a random variable using simulation involves generating a large number of independent samples and computing their average. This sample mean approximates the true mean of the distribution, and the accuracy improves with an increasing number of samples, in accordance with the central limit theorem.

Simulation of Extreme Values

Simulating extreme values, such as the minimum or maximum of a set of random variables, involves generating multiple samples and applying the appropriate function (e.g., min or max) over each set. This technique is often used to assess risk or rare events, and it leverages Monte Carlo methods to estimate probabilities and expectations related to extreme outcomes.

Transcript

00:01 We are asked to simulate the distribution with the pdf shown here using the accept slash reject method with a uniform random variable on 0 to 1.

00:12 So the uniform distribution on 0 to 1, we'll call that g of x, is equal to 1, and x goes from 0 to 1.

00:29 Now for part a, we're asked to find the smallest constant c, such that f at x over g at x is always at most c.

00:45 So f at x over g at x is equal to simply f at x since j at x is 1.

01:04 And this must always be less than c.

01:10 Now the biggest that this function becomes is when x is equal to 0.

01:15 In that case it equals 3 over 2.

01:21 So we will set c equal to 3 over 2.

01:25 That's the smallest constant such that f at x over g at x is always less than c.

01:34 And then in part b we are asked to write a program.

01:36 To simulate values from this distribution.

01:40 Now first recall the steps of the accept -reject method.

01:50 Step 1 is to generate a variant from the function g.

02:02 2 is to generate a uniform variant.

02:14 And then for step 3, so if the uniform variant times the variant from the function g times our constant c is less than or equal to the function at a function at a function at a function y, and this is the function f at y, then we accept x.

02:50 We accept y as x.

02:59 And we keep repeating these three steps until we have the number of simulated values that we want.

03:07 So the r code that can achieve this is shown here.

03:14 So here we are performing step one.

03:17 This is step two, and this is step three.

03:22 If y is accepted as x, then the vector x gets populated with y.

03:27 At the ieth index...

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