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Probability calculations help us understand how likely an event is to occur. By using the cumulative distribution function (CDF) in this exercise, we uncover these likelihoods. CDF is a helpful tool, especially when dealing with continuous random variables. It shows the probability that the random variable is less than or equal to a certain value. In our exercise:

• For part (a), we found that the probability of the random variable being less than or equal to 3 is 1. This means it is certain to happen.
• For part (b), the probability of the random variable being less than or equal to 2 is 0.5. This indicates that there is a 50% chance for this event.
• For part (c), the probability of the variable being between 1 and 2 is calculated to be 0. This means there's no chance of that happening.
• For part (d), we are interested in the probability of the variable being greater than 2, calculated as 0.5 or a 50% chance.

As you can observe, understanding how to read and use a CDF is crucial in performing these calculations.
It gives insight into the behavior of random variables over specific intervals.

Random variables are central in probability and statistics. These variables can take on a range of values based on the outcome of a random event. With our exercise, we work with a random variable, let's call it X. We use the cumulative distribution function to understand which values X can take and the likelihood of each value. There are two main types of random variables:

• Discrete Random Variables: These are countable, like the outcome of rolling a die.
• Continuous Random Variables: These take on values within a range, like measuring someone’s height.

The CDF helps us to view the probability of X being at or below a specific point. In this case, X takes values in intervals between 1 and 3. Here, the possible values determine the different probabilities seen in our solution.

Breaking down problems into step by step solutions can immensely help in understanding. Let's take a look at how we tackled this particular problem:Step 1: We started by interpreting the CDF, understanding how it represents probabilities for variable X's values or intervals.
Step 2: Calculate probabilities for each distinct condition, ensuring each part of the problem is addressed separately.

• For \(X \leq 3\), we saw the total probability is 1.
• Calculation for \(X \leq 2\) depended on the value of the CDF at 2, which is 0.5.
• For the range \(1 \leq X \leq 2\), the probability was determined to be 0, as expected due to the uniform values of CDF at those points.
• Finally, for \(X > 2\), the probability is the complement of \(X \leq 2\), resulting in 0.5.

By visualizing each step, the solution becomes more approachable.
It’s easier to follow what's happening and why each result appears as it does.

In applied statistics, it's essential to understand and utilize probabilities to make data-driven decisions. Learning to analyze functions like the cumulative distribution function gives valuable insights into predicting outcomes.
Applied statistics harnesses these calculations, enabling applications such as:

• Risk Management: Calculating risk probabilities in finance.
• Quality Control: Assuring product quality based on defect probabilities.
• Forecasting: Predicting trends using probability functions.

This exercise introduces the CDF as a tool in applied statistics. With its help, probabilities help inform decisions based on statistical data. We realize that beyond random event prediction, CDF also plays a pivotal role in areas such as operations research, economics, and engineering. Practicing with these calculations strengthens our capability in handling real-world statistical challenges.