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A discrete random variable is a type of variable that can take on a countable number of distinct outcomes. Unlike continuous random variables that can take any value within a range, discrete random variables only have specific possible values. In the context of the provided exercise, the yearly revenue from the product is modeled as a discrete random variable, denoted by \( X \). This is because the revenue can only be one of three distinct amounts: \(10 million, \)5 million, or $1 million.

To fully understand a discrete random variable, think about how it functions similarly to rolling a dice; the outcome can only be one of the fixed numbers on the dice - in this case, the set amounts of revenue. Each outcome is associated with a specific probability, which leads directly into the concept of the Probability Mass Function (PMF).

Revenue outcomes refer to the potential financial results from selling the product, each associated with a different level of success. In this scenario, there are three potential outcomes:

• Very successful: Yearly revenue of $10 million.
• Moderately successful: Yearly revenue of $5 million.
• Unsuccessful: Yearly revenue of $1 million.

These outcomes are critical because they help in understanding the variety of financial states the product might achieve in a year. To make important business decisions, you need to know these outcomes and their probabilities because they allow companies to calculate the expected revenue and assess the risks involved with the product's performance in the market.

Assigning probabilities to the different outcomes is a foundational step in understanding and using statistical models. The probability associated with each revenue outcome represents the likelihood of that particular outcome occurring.

In this example, the probabilities are:

• Very successful with probability 0.3.
• Moderately successful with probability 0.6.
• Unsuccessful with probability 0.1.

These probabilities need to sum up to 1, as the revenue must definitely fall into one of these categories. By assigning these probabilities, you quantify uncertainty and begin to make informed predictions about future performance. This practice is at the heart of probability theory and is crucial for accurate decision-making.

Statistical methods provide engineers with powerful tools for dealing with uncertainty and variability in engineering projects. Probability mass functions, like the one explored in this exercise, are practical applications of these methods. They allow engineers to model random variables and make informed predictions about real-world phenomena, such as product revenue.

When engineers use statistical techniques, it helps them effectively handle risk and make data-driven decisions. For instance, understanding the probability mass function of a product's revenue could guide them in forecasting, resource allocation, or deciding whether to launch a product. Such methods give engineers the basis to systematically and efficiently tackle problems that would otherwise seem unpredictable.

In sum, statistical methods empower engineers by providing a structured approach to uncertainty, thus enhancing their ability to predict outcomes and optimize processes.