Question

A molecule can occupy different levels \(n_{1}, n_{2} \ldots\) with probabilities \(P_{1}, p_{2} \ldots .\) Suppose there are \(N\) such molecules. The probability for finding the successive levels occupicd by \(N_{1}, N_{2, \ldots}\) molecules is given by the multinomial distribution $$ P\left(N_{1}, N_{2}, \ldots\right)=\frac{N !}{N_{1} ! N_{2} ! \ldots} p_{1}^{N_{1}} p_{2}^{N_{2} \cdots} $$

Short answer

To find the probability of a given distribution of molecules across energy levels, substitute the known values into the multinomial formula, ensuring the number of molecules sums up to \(N\) and that the sum of the probabilities is equal to 1.

Step-by-step solution

Understand the Multinomial Distribution

The multinomial distribution is a generalization of the binomial distribution. It models the probabilities of having counts across several categories or levels (in this case, energetic levels occupied by molecules). The probability mass function (PMF) for the multinomial distribution with categories labeled by indices is given by the provided formula and tells us the probability of having exactly \(N_i\) items in category \(i\) (or level \(i\) for molecules), where each item independently falls into category \(i\) with probability \(p_i\).

Identify the Components of the Formula

The formula provided has several components that we need to identify:- \(N!\) is the factorial of the total number of molecules, which represents the total number of ways to arrange \(N\) molecules without any constraints.- \(N_i!\) is the factorial of the number of molecules in level \(i\), representing the number of permutations within that level, which we're not distinguishing between for the probability calculation.- \(p_i^{N_i}\) represents the probability that a molecule will be found at level \(i\), to the power of how many molecules are actually in level \(i\), reflecting independent events of molecules occupying levels.

Apply the Formula to Calculate a Specific Probability

To find the probability of a specific distribution of molecules across energy levels, input the values for \(N\), \(N_1\), \(N_2\), ..., \(p_1\), \(p_2\), ... into the provided formula. Make sure that the sum of all \(N_i\) values equals \(N\), and the sum of all probabilities \(p_i\) equals 1, as these are fundamental properties of the multinomial distribution.

Key concepts

Stochastic Processes

In the world of probability and statistics, a stochastic process is a collection of random variables representing a process that evolves over time subject to certain probabilistic rules. It's a mathematical framework for modeling systems that exhibit random behavior.

Imagine flipping a coin; each flip is a random event, but if we flip the coin multiple times, we start to see a pattern of heads and tails emerging over several flips. This series of coin flips is a simple example of a stochastic process, particularly known as a Bernoulli process, where each event is independent of the previous ones.

Applying this to the original exercise, think of each molecule as an individual coin flip, with their levels representing 'heads' or 'tails.' Although each molecule's position is random, the collective positions of many molecules follow a distinct probability distribution—the multinomial distribution. This is a stochastic process because the overall state of the molecules (across various energy levels) changes each time a molecule 'chooses' a new level based on its probability.

Probability Mass Function

The probability mass function (PMF) is a function that gives us the probability that a discrete random variable is exactly equal to some value. Since it deals with discrete variables, you can think of the PMF as a list that matches each outcome of a random process with its probability of occurring.

In the context of our exercise, each molecule can occupy a distinct energy level, which is a discrete outcome since a molecule cannot be in more than one energy level at the same time. The PMF for the multinomial distribution specifies the probability that any one arrangement of molecules across different energy levels occurs.

Understanding the PMF for Multinomial Distribution

For multinomial distribution, the PMF is not just a simple list; it's a more complex expression that accounts for multiple categories (or energy levels, in our case) and the probability of observing a specific combination of counts across these categories. Remember, not just the individual probabilities are important but also how they come together to form the entire distribution.

Factorial

The concept of factorial is fundamental in combinatorics and probability. It's symbolized by an exclamation point (!) and is defined as the product of all positive integers up to a given positive integer. For instance, the factorial of 5 (denoted as 5!) is the product of all positive integers from 1 to 5, which would be 5 x 4 x 3 x 2 x 1 = 120.

Factorials arise in probability when we're looking at different ways to arrange a set of objects. In the given exercise, 'N!' represents the total number of distinct ways to arrange N molecules without any restrictions. In contrast, 'N_i!' refers to the number of permutations of molecules within the same energy level.

Role of Factorials in the Multinomial Distribution

For the multinomial coefficient, factorials help us normalize the combination. They divide the total number of arrangements by the number of indistinguishable permutations within each category, thus yielding the number of distinguishable outcomes, which is essential for calculating probabilities in a multinomial context.