Problem 18 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose a series of \(n\) independent trials can end in one of three possible outcomes. Let \(k_{1}\) and \(k_{2}\) denote the number of trials that result in outcomes 1 and 2, respectively. Let \(p_{1}\) and \(p_{2}\) denote the probabilities associated with outcomes 1 and 2 . Generalize Theorem \(3.2 .1\) to deduce a formula for the probability of getting \(k_{1}\) and \(k_{2}\) occurrences of outcomes 1 and 2, respectively.

Step-by-Step Solution

Answer

The formula for the probability of getting \(k_{1}\) and \(k_{2}\) occurrences of outcomes 1 and 2, respectively, is \[ P = \frac{n!}{k_{1}! k_{2}! (n - k_{1} - k_{2})!} p_{1}^{k_{1}} p_{2}^{k_{2}} (1 - p_{1} - p_{2})^{n - k_{1} - k_{2}} \].

Step 1: Understanding the problem

In this situation, there are \(n\) trials, and each trial can result in one of three outcomes. Annual occurrences of outcomes 1 and 2 are represented by \(k_{1}\) and \(k_{2}\), respectively, and their respective probabilities are denoted by \(p_{1}\) and \(p_{2}\). The third outcome has \(n - k_{1} - k_{2}\) occurrences and a probability \(1 - p_{1} - p_{2}\).

Step 2: Apply the multinomial theorem

The multinomial theorem, which is an extension of the binomial theorem, is ideal for this scenario because it deals with situations with more than two outcomes. The theorem states that the probability \(P\) will be equal to the formula: \[ P = \frac{n!}{k_{1}! k_{2}! (n - k_{1} - k_{2})!} p_{1}^{k_{1}} p_{2}^{k_{2}} (1 - p_{1} - p_{2})^{n - k_{1} - k_{2}} \]

Step 3: Interpret the results

The derived equation is the desired formula related to Theorem 3.2.1. This equation provides the probability \(P\) of achieving \(k_{1}\) and \(k_{2}\) occurrences of outcomes 1 and 2, respectively, during \(n\) independent trials.

Key Concepts

Probability Theory

Probability theory is the mathematical framework used for calculating the likelihood of outcomes in random processes. In scenarios like having multiple possible outcomes, understanding probability helps in predicting the chance of certain events.

When encountering problems with independent trials, as in the exercise, we focus on the probability of different outcomes occurring across those trials. Here, each trial is independent, meaning the outcome of one trial does not affect another. This is crucial because it lays the foundation for using formulas to determine probabilities when outcomes are distinct from one another.

In the context of the multinomial distribution, which is an extension of the binomial distribution, probability theory is used to give us a precise way to find the likely occurrence of various outcomes. This involves using calculations like factorials and powers of probabilities. This step-by-step approach ensures that we account for all possible outcomes efficiently.

Multinomial Theorem

The multinomial theorem is an extension of the binomial theorem, providing a formula for expanding expressions that have more than two terms. This is especially useful when evaluating the probability of multiple outcomes arising from a series of independent trials.

In our example, the multinomial theorem helps calculate the likelihood of each possible outcome combination in independent trials. The formula has:

• Factorials like \( n! \) which help account for the total number of arrangements.
• Terms \( k_{1}! \) and \( k_{2}! \) to adjust for the occurrence of specific outcomes.
• Powers of probabilities \( p_{1}^{k_{1}} \) and \( p_{2}^{k_{2}} \), which represent the contribution of each outcome's likelihood.

This precise formula ensures that even with three outcomes, represented by specific terms, we can determine the combined probabilities affecting each outcome ratio.
The multinomial theorem is invaluable in probability theory because it equips us to compute probabilities in more complex scenarios.

Independent Trials

Independent trials refer to repeated experiments where the outcome of one trial does not influence the next. This concept is foundational in statistical experimentation and probability calculations.

In the given problem, the trials are independent, allowing us to use the multinomial theorem effectively. This independence is significant because it implies that we don't need to adjust our calculations for correlations between trials, simplifying the mathematics considerably.

Each trial having its own outcome means probabilities can be multiplied as they are independent. For example, the product \( p_{1}^{k_{1}} \) denotes the likelihood of observing a particular number of outcome 1 events across several trials. Understanding independent trials helps us ensure precision and accuracy in probability computations and application of related theorems.