The time required to complete a 3 -h final exam is modeled...

The time required to complete a 3 -h final exam is modeled by the following pdf: $$f(x)=\left\{\begin{array}{cc}{\frac{4}{27} x^{2}(3-x)} & {0 \leq x \leq 3} \\ {0} & {\text { otherwise }}\end{array}\right.$$ Consider simulating values from this distribution using the accept-reject method with a uniform "candidate" distribution on the interval $[0,3] .$ (a) Find the smallest majorization constant $c$ so that $f(x) / g(x) \leq c$ for all $x$ in $[0,3] .[$Hint . [Hint: What is the pdf of the uniform distribution on $[0,3] ? ]$ (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) A professor has 20 students taking her class (lucky professor!). Assume her 20 students' completion times on the final exam can be modeled as 20 independent observations from the above pdf. The professor must stay at the final exam until all 20 students are finished (i.e., until the last student leaves). Use your program in (b) to simulate the rv $L=$ time, in hours, that the professor sits proctoring her final exam to 20 students. Use your simulation to estimate $P(L \geq 35 / 12),$ the probability she will have to stay into the last 5 min of the final exam period.

The time required to complete a 3 -h final exam is modeled by the following pdf:
$$f(x)=\left\{\begin{array}{cc}{\frac{4}{27} x^{2}(3-x)} & {0 \leq x \leq 3} \\ {0} & {\text { otherwise }}\end{array}\right.$$
Consider simulating values from this distribution using the accept-reject method with a uniform "candidate" distribution on the interval $[0,3] .$
(a) Find the smallest majorization constant $c$ so that $f(x) / g(x) \leq c$ for all $x$ in $[0,3] .[$Hint . [Hint: What is the pdf of the uniform distribution on $[0,3] ? ]$
(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) A professor has 20 students taking her class (lucky professor!). Assume her 20 students' completion times on the final exam can be modeled as 20 independent observations from the above pdf. The professor must stay at the final exam until all 20 students are finished (i.e., until the last student leaves). Use your program in (b) to simulate the rv $L=$ time, in hours, that the professor sits proctoring her final exam to 20 students. Use your simulation to estimate $P(L \geq 35 / 12),$ the probability she will have to stay into the last 5 min of the final exam period.

Key Concepts

Order Statistics

Order statistics are statistics obtained from the ordered values of a sample. In many problems, including those involving extreme values, one may be interested in the maximum or minimum of a set of observations. Simulating the order statistics, such as the maximum of independently drawn samples from a target distribution, is important in applications where the behavior of the extremal values is crucial, such as determining waiting times or completion times in various scenarios.

Monte Carlo Simulation

Monte Carlo simulation refers to a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. It is especially useful in estimating probabilities and expectations when analytical solutions are difficult or impossible to derive. In the context of the rejection sampling method, Monte Carlo simulation is used to generate large numbers of random samples to approximate the characteristics of the target distribution.

Majorization Constant

The majorization constant, often denoted as c, is the smallest value such that the target probability density function, when divided by the candidate density function, is bounded above by c for all values in the domain. This constant is critical in the accept-reject method because it ensures that the acceptance probability is always less than or equal to one, making the simulation algorithm valid and efficient.

Rejection Sampling (Accept-Reject Method)

The accept-reject method is a Monte Carlo simulation technique used to generate random samples from a target distribution by using samples from an easier-to-sample candidate distribution. This method involves drawing candidate values and accepting them with a probability that is proportional to the ratio of the target density to a scaled candidate density. It is widely applied when direct sampling from the target distribution is challenging.

Uniform Distribution

The uniform distribution is a probability distribution where all outcomes in a specified interval are equally likely. For simulation purposes, particularly in rejection sampling, using a uniform candidate distribution simplifies the process because it has a constant density over the chosen interval. This property is often exploited to generate candidate values efficiently in Monte Carlo methods.

Probability Density Function (pdf)

A probability density function is a function that describes the relative likelihood for a continuous random variable to take on a given value. The function must be non-negative everywhere and integrate to one over its entire domain. PDFs are used to model the distribution of outcomes in various scenarios and are fundamental in the study of probability and statistics.

Transcript

00:01 We are asked to simulate the distribution with the pdf shown here using the accept -reject method with a uniform candidate distribution on 0 to 3.

00:12 For part a, we are asked to find the majorization constant c.

00:18 This is the constant such that f at x over g at x is always smaller than it.

00:31 Now, g at x is the uniform distribution on 0 to 3, so its pdf is given by 1 over 3.

00:47 Then f at x over g at x is therefore 4 over 9 times x squared times 3 minus x.

01:05 So we want to find the smallest value of c such that this expression is always at most c.

01:15 So in other words we want to find the maximum value that this expression can take.

01:21 So we can use calculus to find that.

01:23 It's where the derivative is equal to 0.

01:39 And if we solve for x, we get two possibilities.

01:42 One is x equal to zero, except that is on the boundary of the domain for x.

01:49 So it is not admissible.

01:50 And the other is x is equal to 2.

02:09 And so what we get is c is equal to 19 over 6.

02:18 Then for part b, we were asked to write a program that simulates values from the distribution.

02:28 So remember the three steps of the process using the accept -reject method are first, we draw a random variable from g, then we draw a random variable from the uniform distribution, the standard uniform distribution.

02:49 And then our acceptance criterion is if the uniform variant times c times the variant drawn from the pdf g of y is less than or equal to f at y, then we accept it.

03:27 And so the code that i've made in r is as follows.

03:33 So we have step one where we generate a random draw from y, which is the uniform distribution on 0 to 3.

03:45 Second step we draw from the standard uniform distribution.

03:50 And 3 we check our acceptance criterion.

03:55 If it meets the criterion, then we accept y into our vector x.

04:02 So that after the program is finished running, x will be a vector of 10 ,000...

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