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In statistics, a discrete random variable is a type of variable that can take on a finite or countable number of distinct values. These values are generally whole numbers, which makes them quite different from continuous random variables that can take on any value within a given range. Understanding discrete random variables is essential in probability theory as they are often used to model and analyze random processes in real-world scenarios.
For example, when you roll a die, the outcome is a discrete random variable because it can only be one of six possible numbers: 1 through 6. In the original exercise, the random variable is represented by the set of numbers {13, 18, 20, 24, 27}. Here, each number corresponds to an event with a specific probability assigned. This is called a probability distribution.
Key points to understand about discrete random variables include:

• They assume specific, distinct values.
• The probability of any event occurring is between 0 and 1.
• The sum of all possible event probabilities is always 1.

The mean of a random variable, often referred to as the expected value, is a measure of central tendency that gives us an idea about the average outcome of a random variable over many trials.
To calculate the mean of a discrete random variable, as shown in the original exercise, you need to multiply each possible value by its corresponding probability and then sum up these products. Mathematically, this is represented by the following formula: \[ \mu = \sum_{i}m_i \times P(m_i) \]
Here, \(m_i\) represents a possible value of the random variable, and \(P(m_i)\) is its probability. In the exercise, the calculation involves values 13, 18, 20, 24, and 27, along with their probabilities to get the mean, \(\mu = 19.76\).
Understanding the mean provides insights into what you can generally expect from the random process over time. It helps to make predictions and informed decisions based on statistical data.

Probability calculation is a fundamental concept in probability theory, which involves determining the likelihood of various outcomes. In the context of discrete random variables, it involves assigning probabilities to all possible outcomes that a random variable can take.
The total probability of all outcomes must equal 1, as indicated by the formula: \[ \sum P(x_i) = 1 \] where \(x_i\) represents individual outcomes.
In the exercise, we calculated different probabilities:

• \(P(18)\), which is directly given as 0.25.
• \(P(X > 18)\), calculated by summing probabilities of outcomes greater than 18: \(P(20) + P(24) + P(27) = 0.53\).
• \(P(X \leq 18)\), which sums probabilities of outcomes 13 and 18: \(P(13) + P(18) = 0.47\).

Calculating probabilities provides a structured way of predicting the chance of occurrence for different scenarios, which is indispensable for performing analyses and making informed decisions in numerous fields like finance, insurance, and research.