91Ó°ÊÓ

A random variable is a fundamental concept in probability theory. Think of it as a function that assigns numerical values to the outcomes of a random phenomenon. In simpler terms, if you roll a die, each face can be represented as a number, with each number being a potential outcome.

In the given exercise, the random variable is represented by \( y \), where each possible value of \( y \) is associated with a probability. These possible values range from -3 to 3 and reflect all the potential outcomes of a particular random process. The corresponding probabilities indicate how likely each outcome is to occur.

• The concept of a random variable helps in transforming real-world processes into mathematical forms, which makes it easier to analyze and understand probabilities.
• Random variables can be either discrete, like in our exercise where values are distinct and countable, or continuous, where values can fall anywhere on a continuum.

By understanding random variables, we are better equipped to analyze and compute probabilities, as they form the backbone of probability distributions like the one showcased in the exercise.

Probability calculations form the essence of understanding how likely an event is to occur. In our exercise, we are asked to perform several calculations using the probability values provided for each outcome of the random variable \(y\).

For calculation purposes:

• **Calculating \(P(y \geq 0)\):** This involves adding the probabilities of all outcomes where \(y\) is greater than or equal to 0. These values are 0, 1, 2, and 3, totaling to a probability of 0.08.
• **Calculating \(P(y \leq 1)\):** Here, we sum up the probabilities from \(y = -3\) to \(y = 1\). This range has a total probability of 0.98.
• **Calculating \(P(|y| \leq 1)\):** This indicates all instances where \(y\) is between -1 and 1, inclusive of both values. Here, when summed, the total probability is 0.15.

These calculations help us concretely understand the likelihood of observing certain values within the distribution and give insights into how these events occur relative to one another.

Step-by-step solutions demystify complex problems by breaking them down into manageable parts. This approach is particularly beneficial in probability exercises, where each step builds on the previous one to arrive at a thorough understanding.

In our exercise, the step-by-step solution process involves:

• **Understanding the Probability Distribution:** Recognizing that the sum of probabilities must equal 1 is crucial. This provides a foundation for accurate calculation and interpretation of probabilities.
• **Sequential Calculations:** Performing each required probability calculation by systematically considering each condition, such as \(y \geq 0\) or \(y^2 > 3\), and summing the corresponding probabilities. This methodical approach ensures no steps are skipped or overlooked.
• **Validation and Cross-Checking:** After computation, reviewing calculations to confirm that they align with the premise that the total probability remains consistent at 1. This step assures us of the reliability of our solution.

By using a step-by-step approach, we greatly enhance comprehensive problem-solving skills, making it easier to tackle even more challenging probability problems in the future.