Problem 75 - Calculus for Biology and Medicine

Question

Suppose the waiting time for the first success in an experiment is geometrically distributed with mean \(1 / p\). (a) Find the probability that the first success occurs on the \(k\) th trial. (b) The experiment is repeated after the first success. Assume that the waiting time for the second success has the same distribution as the waiting time for the first success. Find the probability mass function for the distribution of the second success.

Step-by-Step Solution

Answer

(a) \(P(X = k) = (1-p)^{k-1}p\). (b) PMF for second success is \(P(Y = k) = (1-p)^{k-1}p\).

Step 1: Understand the Geometric Distribution

The geometric distribution models the number of trials needed to get the first success in a series of independent and identical Bernoulli trials. If the mean is given by \(1/p\), where \(p\) is the probability of success in each trial, it indicates that each trial is independent and has the same success probability \(p\).

Step 2: Find Probability of First Success on k-th Trial

The probability that the first success occurs on the \(k\)th trial for a geometric distribution is given by the formula: \[P(X = k) = (1-p)^{k-1} \cdot p\].Here, \( (1-p)^{k-1} \) is the probability of \(k-1\) failures before the first success, and \(p\) is the probability of success on the \(k\)th trial.

Step 3: Analyze Repeating the Experiment for Second Success

Since the trials are independent and the success probability remains \(p\), the waiting time for the second success has the same distribution as that of the first success. Therefore, the probability mass function (PMF) for the second success occurring on the \(k\)th trial is also geometric.

Step 4: Write PMF for Distribution of Second Success

The probability that the second success occurs on the \(k\)th trial is essentially another set of geometric trials starting from 0, so the PMF remains: \[P(Y = k) = (1-p)^{k-1} \cdot p\].This is the same geometric distribution because each set of trials are independent with the same probability \(p\).

Key Concepts

Probability Mass Function

The Probability Mass Function (PMF) is a fundamental tool in understanding discrete probability distributions. For a geometric distribution, the PMF helps us determine the likelihood of the first success occurring at a particular trial number.

In the context of a geometric distribution, the PMF is defined mathematically as:\[P(X = k) = (1-p)^{k-1} \cdot p\]Here:

• \(X = k\) represents the event that the first success happens on the \(k\)th trial.
• \( (1-p)^{k-1} \) accounts for the \(k-1\) failures preceding the first success.
• \(p\) is the probability of success occurring on the \(k\)th trial.

The PMF provides a complete representation of the distribution, seamlessly illustrating its "memoryless" property — the probability distribution remains unchanged no matter how many trials have already been conducted, as long as each trial is independent.

Bernoulli Trials

Bernoulli Trials are the building blocks of various probability distributions, including the geometric distribution. Each trial has two possible outcomes: success or failure.

In a sequence of Bernoulli Trials, several important conditions hold true:

• Each trial is independent of the others.
• The probability of success, \(p\), remains constant throughout all trials.
• A single trial has binary outcomes: either it is a success or a failure.

The simplicity of Bernoulli Trials allows for the straightforward derivation of the geometric distribution's properties. Since every trial is independent, the outcome of any particular trial does not influence others, making it convenient to model phenomena like waiting times until the first success or repeated experiments.

Independent Trials

The concept of Independent Trials is crucial in probability theory and underlies many statistical models, including the geometric distribution. When trials are independent, the outcome of one does not affect the outcomes of others.

In applications involving independent trials:

• The probability of success in one trial is unaffected by any previous or future outcomes.
• Each trial can be modeled as a separate experiment with its own set of probabilities.
• This simplification allows for predictions and calculations based on the assumption that each trial resets the statistical scenario.

Such independence is also the reason the geometric distribution exhibits the "memoryless" property. Even when starting over or repeating experiments, the probability distribution stays the same. This inherent characteristic facilitates modeling of scenarios where events are restarted at each non-success.