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A geometric random variable, often denoted as \(X\), is a type of discrete random variable. It represents the number of trials needed to get the first success in a sequence of Bernoulli trials.
Bernoulli trials are simple experiments where there are only two possible outcomes: success or failure. For example, flipping a coin until you get heads can be thought of as a series of Bernoulli trials.
The key parameter for a geometric random variable is \(p\), the probability of success on any given trial. If \(p\) is the probability of success, then \(1-p\) is the probability of failure.
The geometric random variable has the following distinctive feature:

• Each trial is independent of others.
• The probability of success and failure remains constant across trials.

An important characteristic is its memoryless property. It means that the probability of success on any given trial is independent of past trials.

The probability mass function (PMF) of a geometric random variable provides the probabilities of different outcomes. Specifically, it tells us the probability that the first success occurs on the k-th trial.
Mathematically, the PMF is expressed as: \[ \text{Pr}(X = k) = (1-p)^{k-1} p \] Here, \(k\) is the trial number, \(p\) is the probability of success, and \((1-p)^{k-1}\) accounts for the previous \(k-1\) failures.
This formula captures the essence of geometric distribution: the probability decreases exponentially with each trial.
Some key points to remember about the PMF include:

• It only applies to integer values of \(k \ge 1\).
• Summing the PMF over all possible values of \(k\) yields 1, which aligns with the total probability rule.

Understanding the PMF is crucial for solving problems involving geometric random variables, as it forms the basis for further calculations.

A geometric series is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio.
In our context, the expressions we deal with in the problem form a geometric series. The terms are: \[ p, (1-p)p, (1-p)^2 p, \dots, (1-p)^{n-2} p \]
For a geometric series where the first term is \(a\) and the common ratio is \(r\), the sum of the first \(n-1\) terms is given by: \[ S_{n-1} = a \frac{1-r^{n-1}}{1-r} \]
Applying this to our expression, we have: \[ S_{n-1} = p \frac{1-(1-p)^{n-1}}{1-(1-p)} \] Substituting \(a=p\) and \(r=1-p\)
This formula helps us compute the cumulative probability \(\text{Pr}(X < n)\) by summing the probabilities of all outcomes from 1 to \(n-1\).
Geometric series are fundamental in understanding the behavior and computations involving geometric distributions.