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When we talk about the Cumulative Distribution Function (CDF), we are referring to a function that provides the probability of a random variable being less than or equal to a particular value. In simpler terms, it maps every possible outcome to a probability indicating how likely the random variable is to be below or at that outcome.
For a continuous random variable like in the given exercise, the CDF is essential because it accumulates the probabilities up to a specific point by integrating the Probability Density Function (PDF).
The CDF can be given by:

• For values less than the minimum of the range: 0.
• For values within the range: the result of the integral of the PDF.
• For values greater than the maximum of the range: 1.

To achieve this, you calculate the integral of the PDF from the start of the range to your current value. This gives you the CDF, helping you understand the distribution of the random variable fully.

A pivotal quantity is a unique concept in statistics. It is a function that combines your sample data and some unknown parameters, but crucially, it has a known distribution that is independent of those unknown parameters. This characteristic makes pivotal quantities incredibly useful in constructing confidence intervals.
In the context of the exercise, \( \frac{Y}{\theta} \) is shown to be a pivotal quantity. By transforming \( Y \) into \( U = \frac{Y}{\theta} \), we see that \( U \) has a distribution that does not depend on \( \theta \). Since pivotal quantities remove dependence on unknown variables, they are fundamental tools for statistical estimation and hypothesis testing. Here, they allow us to derive estimates for other parameters without being skewed by unknown variables.

Confidence intervals provide us with a range of values in which we believe an unknown parameter, like \( \theta \), lies with a certain degree of confidence. They combine our sample data analysis and pivotal quantities to find this plausible range.
Typically, confidence intervals are expressed in terms of a confidence level, such as 90%, which indicates that 90% of similarly constructed intervals will contain the parameter. To construct a confidence interval using a pivotal quantity, you solve the distribution of the pivotal quantity to find its critical values that correspond to your desired confidence level.
For instance, in our example, the 90% lower confidence limit gives you a sense of how low \( \theta \) might be, backed up by statistical theory. This approach offers a statistically sound way to make inferences about population parameters, even when they cannot be directly observed.

A continuous random variable is a variable that can take on an infinite number of possible values within a given range. Unlike discrete random variables, which have distinct and separate values, continuous random variables are associated with probability density functions (PDFs) rather than probability mass functions (PMFs).
In many real-world scenarios, such as measuring weight, time, or in our particular exercise, continuous random variables allow us to model these measurements smoothly across their possible outcomes. The key characteristic is that, while individual outcomes have zero probability (since their exact probability cannot be pinpointed), the probability is distributed across a range of values. This means that probability is defined over intervals.
Continuous random variables provide the foundational knowledge for understanding more complex statistical concepts, such as probability density functions, cumulative distribution functions, and how these are used to analyze statistical data effectively.