Question

Suppose \(n\) balls having weights \(w_{1}, w_{2}, \ldots, w_{n}\) are in an urn. These balls are sequentially removed in the following manner: At each selection, a given ball in the urn is chosen with a probability equal to its weight divided by the sum of the weights of the other balls that are still in the urn. Let \(I_{1}, I_{2}, \ldots, I_{n}\) denote the order in which the balls are removed-thus \(I_{1}, \ldots, I_{n}\) is a random permutation with weights. (a) Give a method for simulating \(I_{1}, \ldots, I_{n}\). (b) Let \(X_{i}\) be independent exponentials with rates \(w_{i}, i=1, \ldots, n .\) Explain how \(X_{i}\) can be utilized to simulate \(I_{1}, \ldots, I_{n}\).

Short answer

(a) To simulate the sequence \(I_1, \ldots, I_n\), perform the following steps: 1. Create an urn with balls labeled 1 to n with weights \(w_1,...,w_n\). 2. Calculate the probability of choosing each ball as \(P(i) = \frac{w_i}{\sum_{j\in \text{urn}} w_j}\). 3. Choose a ball based on the probability distribution and remove it from the urn. 4. Repeat steps 2-3 until the urn is empty. (b) To simulate the sequence \(I_1, \ldots, I_n\) using exponential random variables \(X_1,...,X_n\): 1. Generate exponential random variables with rates \(w_1,...,w_n\). 2. Sort the variables in ascending order and record their indices, which represent the sequence \(I_1, \ldots, I_n\).

Step-by-step solution

Part (a) - Simulating ball removal based on weights.

1. Initialize setup: Create an urn with balls labeled 1 to n and their corresponding weights \(w_1,...,w_n\). Initialize an empty list for the sequence \(I_1,...,I_n\). 2. Calculate probabilities: For each ball i in the urn, calculate the probability of choosing that ball as \(P(i) = \frac{w_i}{\sum_{j\in \text{urn}} w_j}\). Normalize the probabilities to ensure they add up to 1 (due to possible rounding errors). 3. Choose a ball based on probability: Using the calculated probabilities, choose a ball from the urn based on the probability distribution. Let that ball be ball k. 4. Remove ball k from the urn: Remove ball k from the urn and append its label to the end of the list representing the sequence \(I_1,...,I_n\). 5. Repeat steps 2-4 until the urn is empty: Continue this process until all balls have been removed from the urn. The resulting list represents the simulated sequence \(I_1,...,I_n\).

Part (b) - Utilizing exponential random variables for simulation.

1. Simulate exponential random variables: Independently generate n exponential random variables \(X_1,...,X_n\), each with rate \(w_1,...,w_n\). You can do this using a random number generator and transformation of the uniform random variables. 2. Determine the order based on random variables: Sort the exponential random variables in ascending order (from the smallest to the largest value) and record their indices. The sorted indices represent the sequence \(I_1,...,I_n\). 3. Interpretation: The interpretation of this method is that the exponential random variables \(X_1,...,X_n\) represent the waiting times for the balls to be removed from the urn. The smaller the value of the exponential variable, the sooner the ball with that index will be removed, and hence smaller values correspond to earlier positions in the sequence \(I_1,...,I_n\).

Key concepts

Probability Simulation

Simulating probabilities is much like a digital replication of a randomized, real-world process. Imagine you're playing a game of chance with balls of different weights – it's a bit like trying to guess which one will come out of a bag first. The key is the randomness and the weighted nature of the choices.

In our textbook exercise, each ball's chance of being picked is in proportion to its weight compared to the remaining balls. This selection mimics how probabilities work in real life, where events with higher 'weight' or likelihood occur more frequently. To simulate this digitally, we perform calculations that consider the relative weights and use a random number generator to mimic the randomness inherent in probabilistic outcomes.

The improved method provided in the step-by-step solution suggests uniformly distributing the probabilities first. This is important because the random number generator operates under the assumption of an even probability distribution, and so we must adjust for this to accurately represent weighted random events, thereby teaching the computer to 'understand' the skewed nature of our urn game.

Exponential Random Variables

When we talk about exponential random variables in the context of probability and statistics, we're dealing with a special type of continuous probability distribution. These variables are compelling in simulating 'time until event' situations because they have the memoryless property – the passage of time does not affect the probability of the event occurring.

How does this relate to simulating our urn game? Well, imagine that each ball has to wait a certain amount of time before it can be selected – and this wait time follows an exponential distribution, with the 'rate' tied to the ball's weight. A heavier ball means a shorter expected wait, thus a higher chance of it being selected soonest. By simulating these wait times for each ball and arranging them from the shortest to the longest, we effectively simulate the order in which the balls are drawn – offering an alternative method to directly simulating weighted selections, and adding a temporal dimension to the randomness.

The solution suggests that by sorting these simulated wait times, we get our sequence – an elegant workaround to simulating our game using principles from a seemingly unrelated area of probability theory.

Weighted Selection

Weighted selection is a technique in probability simulations where choices are not equally likely but instead have different chances of being selected based on assigned weights. It is akin to having a lottery with tickets of different sizes – the larger the ticket, the higher the chance it has to win.

In the context of our urn game, each ball's weight influences its likelihood of being chosen. Unlike a fair coin toss, where the outcomes are equally probable, weighted selection skews the odds based on assigned weights. So how do we actually simulate this skewed random pick? We use a mathematical approach to ensure that the probability of selecting each ball is proportional to its weight. Once the probabilities are accurately calculated and normalized, we can then use a random number generator to pick a 'winner' based on these probabilities, and then repeat until we've exhausted all our options.

This nuanced method requires meticulous attention to detail – particularly in ensuring that all probabilities are properly normalized and that we account for the diminishing pool of options as each ball is selected. By doing so, we ensure that our simulation reflects the intended probabilities at each step, offering students a practical insight into how weighted randomness operates.