Question

Consider the Ehrenfest urn model in which \(M\) molecules are distributed between two urns, and at each time point one of the molecules is chosen at random and is then removed from its urn and placed in the other one. Let \(X_{n}\) denote the number of molecules in urn 1 after the \(n\) th switch and let \(\mu_{n}=E\left[X_{n}\right]\). Show that (a) \(\mu_{n+1}=1+(1-2 / M) \mu_{n}\). (b) Use (a) to prove that $$ \mu_{n}=\frac{M}{2}+\left(\frac{M-2}{M}\right)^{n}\left(E\left[X_{0}\right]-\frac{M}{2}\right) $$

Short answer

The solution to the problem can be found using conditional expectation, the properties of expectation in probability, and solving a recurrence relation. The key results are: (a) The expected number of molecules in urn 1 after the \((n+1)\)th switch is given by the following relation: $$ \mu_{n+1}=1+(1-\frac{2}{M})\mu_{n} $$ (b) The general expression for the expected number of molecules in urn 1 after the \(n\)th switch is: $$ \mu_n = \frac{M}{2} + \left(\frac{M-2}{M}\right)^n\left(E[X_0] - \frac{M}{2}\right) $$

Step-by-step solution

For each molecule, the probability of being chosen at a given time is \(\frac{1}{M}\). We can find the conditional expectation of the number of molecules in urn 1 after the \((n+1)\)th switch, given the number of molecules after the \(n\)th switch: $$ E[X_{n+1} | X_n] = P(X_{n+1}=X_n+1 | X_n) + P(X_{n+1}=X_n-1 | X_n) $$ The probability of a molecule from urn 2 being chosen and placed in urn 1 is \(\frac{M - X_n}{M}\), which increases the number of molecules in urn 1 by 1. The probability of a molecule being chosen from urn 1 and placed in urn 2 is \(\frac{X_n}{M}\), which decreases the number of molecules in urn 1 by 1. So, we have: $$ E[X_{n+1} | X_n] = \frac{M - X_n}{M}(X_n + 1) + \frac{X_n}{M}(X_n - 1) $$ #Step 2: Find the Unconditional Expectation#

To find the unconditional expectation, we take the expectation of the conditional expectation calculated in Step 1: $$ E[X_{n+1}] = E\left[E[X_{n+1} | X_n]\right] $$ Now we substitute the expression of \(E[X_{n+1} | X_n]\) obtained in Step 1: $$ E[X_{n+1}] = E\left[\frac{M - X_n}{M}(X_n + 1) + \frac{X_n}{M}(X_n - 1)\right] $$ Now use the linearity of expectation: $$ E[X_{n+1}] = \frac{1}{M} E[(M-X_n)(X_n + 1)] + \frac{1}{M} E[X_n(X_n - 1)] $$ #Step 3: Simplify the Expectation#

Expand the terms and simplify: $$ E[X_{n+1}] = \frac{1}{M}(E[M X_n - X_n^2 + M - X_n]) + \frac{1}{M}(E[X_n^2 - X_n]) $$ Using the linearity of expectation again and noticing that \(E[M]=M\), we have: $$ E[X_{n+1}] = \frac{1}{M}(M E[X_n] - E[X_n^2] + M^2 - M E[X_n]) + \frac{1}{M}(E[X_n^2] - E[X_n]) $$ Now, group the terms with \(E[X_n]\) and \(E[X_n^2]\), and simplify: $$ E[X_{n+1}] = 1 - \frac{2}{M} E[X_n] + E[X_n] $$ Thus, we obtain the desired expression for part (a): $$ \mu_{n+1}=1+(1-\frac{2}{M})\mu_{n} $$ #Step 4: Solving the Recurrence Relation#

We have the following recurrence relation for the expected number of molecules in urn 1: $$ \mu_{n+1}=1+(1-\frac{2}{M})\mu_{n} $$ We can solve this by iteration, starting with the initial condition \(\mu_0=E[X_0]\): $$ \mu_n = \left(1-\frac{2}{M}\right)^n \mu_0 + \sum_{k=0}^{n-1}\left(1-\frac{2}{M}\right)^k $$ Now, we can use the formula for the sum of a geometric series: $$ \sum_{k=0}^{n-1}\left(1-\frac{2}{M}\right)^k = \frac{1-\left(1-\frac{2}{M}\right)^n}{\frac{2}{M}} $$ Plugging this expression back into the formula for \(\mu_n\), and simplifying, we get: $$ \mu_n = \frac{M}{2} + \left(\frac{M-2}{M}\right)^n\left(E[X_0] - \frac{M}{2}\right) $$ This is the expression for \(\mu_n\) that we needed to prove in part (b).

Key concepts

Probability Models

A probability model is a mathematical representation of a random process that allows us to predict the likelihood of various outcomes. In the context of the Ehrenfest urn model, the probability model used demonstrates the random selection of molecules and their transfer between two urns. It hinges on the fundamental principles of probability theory, particularly the concepts of random variables and expectations.

In this scenario, our random variable is the number of molecules in one of the urns after each switch, denoted by \(X_{n}\), and the probability model must consider all the possible ways a molecule could be chosen and moved. By establishing the probability of each molecule’s movement between the urns, we can set up expectations, such as the expected number of molecules in the first urn after the nth switch, \(E[X_{n}]\), which is key to understanding the evolution of the system over time.

Understanding probability models and their application to real-world scenarios, like the Ehrenfest model, allows for critical insights into how randomness influences system dynamics. For students, grasping the probability model involved in this exercise is paramount to solving the problem, and it serves as a foundational skill for tackling similar probabilistic scenarios.

Conditional Expectation

Conditional expectation is a crucial concept in the field of probability, referring to the expected value of a random variable given that certain conditions are met. In the Ehrenfest urn model, the conditional expectation \(E[X_{n+1} | X_n]\) represents the expected number of molecules in urn 1 after the \((n+1)\)th switch, given the number of molecules after the \(n\)th switch.

The step-by-step solution uses the idea of conditional expectation to show how the expected number of molecules changes after each switch. This involves calculating the probability of each possible outcome of the next switch (either gaining or losing a molecule) and the corresponding new number of molecules in urn 1. The calculation of conditional expectation demonstrates how knowing the current state of the system (the number of molecules in urn 1 at the \(n\)th switch) allows us to make an informed prediction about the next state.

For students, understanding conditional expectation is important because it highlights how the probability of future events can depend on the current state of a system. This concept is widely applicable beyond the Ehrenfest urn model and appears in various fields including economics, engineering, and statistics.

Recurrence Relations

Recurrence relations are equations that define sequences iteratively, where subsequent terms are defined based on preceding ones. In the Ehrenfest urn model, we encounter a recurrence relation when predicting the future number of molecules in the urn based on previous observations. As the step-by-step solution illustrates, the expected number of molecules \(\mu_{n+1}\) can be found using the relation \(\mu_{n+1}=1+(1-\frac{2}{M})\mu_{n}\).

What makes recurrence relations particularly useful in probability and other areas of mathematics is their capacity to describe many complex systems simply and elegantly. By solving the recurrence relation given in the problem, we can express the expected number of molecules after any number of switches. This is not only a solution to the problem but also a general formula that can predict the system's behavior under different initial conditions.

For students, mastering recurrence relations is a tool that enables the resolution of problems that may otherwise appear intractable. It is essential in breaking down complex, iterative processes into manageable steps that can be calculated and understood. Learning to solve recurrence relations can thus open doors to advanced methods in solving mathematical, computational, and theoretical problems.