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A discrete random variable is a type of random variable that can take on only specific, distinct values within a given range. In this context, the variable is often used to describe outcomes in scenarios where only integer values make sense.
For the exercise given, let’s denote the discrete random variable as \(y\), which represents the number of broken eggs in a carton of a dozen eggs. Here, \(y\) can only take on values like 0, 1, 2, 3, or 4, because you can't break a fraction of an egg in discrete terms.

Discrete random variables are crucial because they help in modeling different real-world phenomena accurately. By representing specific, countable outcomes, they allow us to use probability distributions to predict events like the one described in the problem. Such applications include quality control in manufacturing or predicting the occurrence of certain events in given trials.

A Probability Mass Function (PMF) is an essential concept in probability theory. It gives the probability that a discrete random variable is exactly equal to some value. The PMF is denoted as \(p(y)\) for a discrete variable \(y\).
In the context of the exercise, the PMF is used to describe the probability distribution of broken eggs in a carton. For example, \(p(1) = 0.20\) indicates that the probability of finding exactly one broken egg in a carton is 20%.

• The PMF fulfills one key rule: the sum of all probabilities \(p(y)\) must equal 1. This is because one of the possible outcomes must surely happen.
• In the given example, notice how probabilities add up to 1 once we find \(p(4)\).

Keep in mind the PMF provides probabilities for specific values of \(y\), making it different from continuous distributions, where probabilities are defined over ranges of values.

The Cumulative Distribution Function (CDF) for a discrete random variable represents the probability that the variable will take a value less than or equal to a certain number. Notated as \(P(y \leq k)\), it sums the probabilities given by the PMF for all values up to \(k\).
For the egg carton problem, calculating \(P(y \leq 2)\) involves summing up the probabilities for \(y = 0\), \(1\), and \(2\), resulting in a total probability of 0.95 or 95%. This indicates that there is a 95% chance that the carton contains 2 or fewer broken eggs.

The CDF is crucial for understanding how probabilities accumulate as you move through the range of a variable. It helps to answer questions about the likelihood of an event happening at or below a certain threshold compared to PMF, which provides probabilities for exact values.