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In statistics, the Probability Mass Function (PMF) is a crucial concept, especially when dealing with discrete random variables like those found in a geometric distribution. It provides the probabilities of different possible outcomes. For a geometrically distributed variable, the formula for the PMF is given by:

• \( P(Y = k) = (1-p)^{k-1} p \)

This equation tells us that for a random variable \( Y \), which follows a geometric distribution, the probability that \( Y \) takes a specific value \( k \) is a product of two components:

• \( (1-p)^{k-1} \): Represents the probability of having \( k-1 \) failures before the first success.
• \( p \): Represents the probability of success on the \( k^{th} \) trial.

The parameter \( p \) denotes the success probability in each trial, while \( 1-p \) is the failure probability. Understanding the PMF helps in recognizing patterns in data and predicting probabilities of different outcomes in a sequence of Bernoulli trials.

An infinite series is the sum of an infinite sequence of numbers. It is a foundational topic in calculus and has expansive applications in mathematics. In our geometric distribution example, dealing with odd integer outcomes involves summing probabilities over infinitely many terms. The series in question is:

• \( \sum_{n=1}^{\infty} (1-p)^{2n-2} p \)

By understanding infinite series, we realize that even though individual probabilities for each odd integer diminish, the sum of these probabilities can converge to a specific value. In this case, we transform the series so that it becomes a more manageable geometric series.
The art of working with infinite series involves recognizing patterns and simplifying expressions. This helps in finding closed-form solutions, which are essential for determining summed probabilities succinctly.

A geometric series is an infinite series in which each term after the first is formed by multiplying the previous one by a fixed, non-zero number called the common ratio. The general form of a geometric series is:

• \( a + ar + ar^2 + ar^3 + \ldots \)
• Sum \( = \frac{a}{1-r} \), for \( |r| < 1 \)

In our situation, after recognizing the pattern of terms \( q^{2n-2} \) in the series, it matches the template of a geometric series with:

• First term \( a = 1 \) (once adjusted for \( n = 0 \))
• Common ratio \( r = q^2 \)

This allows us to sum it easily as \( \frac{1}{1-q^2} \), providing the critical element needed to solve our original probability problem. Geometric series simplify the calculation by reducing complex infinite sums to straightforward formulas.