Problem 76 - Calculus for Biology and Medicine

Question

A Bernoulli experiment with probability of success \(p\) is repeated until the \(n\) th success. Assume that each trial is independent of all others. Find the probability mass function of the distribution of the \(n\) th success. (This distribution is called the negative binomial distribution.)

Step-by-Step Solution

Answer

The PMF of the \(n\)th success in a negative binomial distribution is \(P(X = x) = \binom{x-1}{n-1} p^{n} (1-p)^{x-n}\) for \(x = n, n+1, n+2, \ldots\).

Step 1: Understand the Problem Statement

In this exercise, we have a Bernoulli trial, which is a simple random experiment with two possible outcomes: success (with probability \(p\)) and failure (with probability \(1-p\)). We continue these trials independently until we achieve the \(n\)th success. We need to find the probability mass function (PMF) of the random variable that represents the number of trials required to achieve this \(n\)th success, known as the negative binomial distribution.

Step 2: Define the Random Variable

Let \(X\) be the random variable representing the number of trials needed to achieve the \(n\)th success. For the \(n\)th success to occur on the \(x\)th trial, the first \(x-1\) trials must contain exactly \(n-1\) successes and \(x-n\) failures. The \(x\)th trial must be a success.

Step 3: Identify the Combination of Events

The sequence consists of \(n-1\) successes in the first \(x-1\) trials, followed by a success on the \(x\)th trial. The number of unique ways to arrange \(n-1\) successes in \(x-1\) trials is given by the binomial coefficient \(\binom{x-1}{n-1}\).

Step 4: Determine Probability of Each Arrangement

Each arrangement of trials (having \(n-1\) successes and \(x-n\) failures) contributes to the probability as follows: the probability of any specific arrangement is \(p^{n-1}\) for the successes, \((1-p)^{x-n}\) for the failures, and \(p\) for the last success on the \(x\)th trial. Thus, the probability for an arrangement is \(p^{n}(1-p)^{x-n}\).

Step 5: Formulate the Probability Mass Function

By combining the coefficients and probabilities calculated in the previous steps, the PMF for the negative binomial distribution is given by:\[ P(X = x) = \binom{x-1}{n-1} p^{n} (1-p)^{x-n} \]for \(x = n, n+1, n+2, \ldots\). This equation represents the probability of needing \(x\) trials to get \(n\) successes.

Key Concepts

Bernoulli Trial

A Bernoulli trial is a fundamental concept in probability theory, named after the Swiss mathematician, Jacob Bernoulli. It's essentially a simple random experiment with only two possible outcomes: success and failure.

• A success occurs with probability \( p \).
• A failure occurs with probability \( 1-p \).

Each trial is independent, meaning the outcome of one trial does not influence another. This is crucial in forming standard probability distributions like the binomial or negative binomial distributions.
The Bernoulli trial is the building block for much more complex probability scenarios. In our exercise, we are conducting repeated Bernoulli trials until achieving a fixed number of successful outcomes, specifically the \( n \)th success. This extends the simple concept of a single Bernoulli trial into a broader sequence of trials, maintaining independence throughout them.

Probability Mass Function

A probability mass function (PMF) describes the probability distribution of a discrete random variable. It specifies the probability that a random variable is exactly equal to some value.
In the case of the negative binomial distribution discussed in the exercise, the PMF provides the probability that \( X = x \), where \( X \) is the random variable representing the number of trials needed to achieve the \( n \)th success.

• For a negative binomial distribution, the PMF is given by: \[ P(X = x) = \binom{x-1}{n-1} p^{n} (1-p)^{x-n} \]
• Here, \( \binom{x-1}{n-1} \) is the number of combinations for choosing \( n-1 \) successes from \( x-1 \) trials.
• \( p^n \) is the probability of the \( n \) successes, and \( (1-p)^{x-n} \) accounts for the accompanying failures needed by the \( x-th \) trial.

The PMF helps us understand the likelihood of different outcomes, making it a vital tool in probability theory. This function effectively answers 'How likely is it that exactly \( x \) trials are needed to achieve \( n \) successes?'.

Binomial Coefficient

The binomial coefficient \( \binom{n}{k} \) is a mathematical expression that describes how many ways you can choose \( k \) successes from \( n \) trials, without regard to order. It's a central part of combinatorics and is used extensively in probability.

• The formula for the binomial coefficient is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
• Here, \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).

In the context of our exercise on the negative binomial distribution, we use the coefficient \( \binom{x-1}{n-1} \). This counts the distinct ways to arrange \( n-1 \) successes in \( x-1 \) trials. It involves selecting positions for the successes from the \( x-1 \) trials.
Binomial coefficients simplify calculations for problems involving discrete probability distributions and permutations, making them indispensable in probability and statistics.