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In probability theory, a discrete random variable is a type of random variable that can take on a finite or countably infinite set of distinct values. These values are typically whole numbers or entities that can be listed out individually. Take, for example, the number of years employees stay at a company as described in the exercise. Each year is a distinct, countable value, making the number of years a discrete random variable.

A discrete random variable is characterized by its probability distribution, which assigns a probability to each possible value the variable can take. This probability distribution can be visualized in tables, graphs, or equations. It provides us with the likelihood of each potential outcome, making it a powerful tool for predicting and analyzing events. Discrete random variables differ from continuous random variables, which can take on an infinite number of values within a range, such as temperature or time.

A fundamental property of probability distributions for discrete random variables is that the sum of all probabilities equals 1. This rule reflects the basic principle that one of the possible outcomes must happen.

In the context of the exercise provided, the probability distribution of the number of years employees stay at a company lists probabilities for different durations. These probabilities, when added together, must total 1 to account for all possible scenarios. For instance, if there are probabilities for staying 1, 2, 3, 4, and 5 years, their total must be 1:

• The sum of all probabilities ensures that something in the defined set of outcomes will occur.
• It confirms the distribution is correctly specified and complete.

Checking that probabilities sum to 1 is critical in assessing the validity of any probability distribution.

Attrition rate analysis involves examining how long employees remain with a company before they leave for various reasons. This analysis is essential for businesses, as it helps them understand employee retention and the factors influencing turnover.

Attrition rate is often expressed as a percentage or derived from a probability distribution model, like the one given in the exercise, where probabilities are assigned to different lengths of service.

Key benefits of attrition rate analysis include:

• Identifying trends and patterns in employee turnover.
• Assessing the effectiveness of hiring and onboarding strategies.
• Predicting future staffing needs and costs related to hiring and training.

By analyzing the distribution of time employees spend with the company, organizations can make informed decisions about improving retention strategies and enhancing workplace satisfaction.