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The concept of a Probability Mass Function (PMF) is fundamental in probability theory, especially when dealing with discrete random variables. A PMF gives the probability that a discrete random variable is exactly equal to some value. For example, in our exercise, the PMF for the number of defects per yard in a certain fabric is given by:

• \( p(y) = \left( \frac{1}{2} \right)^{y+1} \), where \( y = 0, 1, 2, \dots \)

This formula tells us how likely it is to find \( y \) defects in one yard of fabric. Essentially, the higher the number of defects, the smaller the probability, as seen by the exponential nature of the function with base \( \frac{1}{2} \). This function declines rapidly as \( y \) increases, which means the probability of observing a high number of defects is quite low.

Discrete random variables are variables that can take on a countable number of distinct values. In our specific exercise, the variable \( Y \) represents the number of defects found in a yard of fabric.
Discrete random variables are contrasted with continuous random variables, which can take on any value within a given range. Examples of discrete random variables include the number of heads obtained when flipping a coin several times or the total number of students present in a class on a given day.
Calculating expected values or probabilities, such as the PMF, for discrete variables is straightforward because they only take on specific values (e.g., 0, 1, 2, etc.). The exercise makes use of this by calculating the expected number using a given PMF.

An infinite series is a series that continues indefinitely. In mathematics, an infinite series is typically written in the form

• \( \sum_{n=0}^{\infty} a_n \), where the \( a_n \) are terms of the series.

In our example, the calculations involved the sum of an infinite geometric series to find the expected value of number of defects per yard.
The key to solving this infinite series was realizing that each term followed the geometric pattern \( y (1/2)^y \). Utilizing the known formula for the sum of an infinite series, namely

• \( \sum_{y=0}^{\infty} y x^y = \frac{x}{(1-x)^2} \)

where \( |x| < 1 \), allowed us to evaluate it effectively.
This allowed substitutions led to simpler calculations, establishing the expected value of one defect per yard in the given fabric scenario.

Defects in fabric can arise due to various causes, such as issues in the manufacturing process or the material itself. In quality control and production, these defects are usually measured and tracked due to their economic impact.
The exercise deals with quantifying these defects statistically, using the concept of expected values.
By determining the expected number of defects through probabilistic models, companies can better predict and manage the quality of their produced fabrics.

• Reducing defects leads to better product quality.
• It also reduces waste and the potential for product recalls or returns.
• Understanding the statistics behind defects helps in implementing more effective quality control measures.

This emphasis on calculating and projecting defects involves using probability theories and series to derive meaningful insights from data about product performance.