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A random variable is a measurable function from a set of possible outcomes to a measurable space. In simple terms, it's a variable that can take on different values, each with a certain probability. For instance, in the context of this exercise, the random variable is the proportion of impurities \( Y \) in ore samples. This \( Y \) is characterized by a probability distribution, defined by a density function \( f(y) \).The idea is that every outcome of the process of picking an ore sample has a certain likelihood, described by \( f(y) \), which provides the probability of the impurity level being within a particular interval. For \( Y \) in the given exercise, the density function is quadratic, which means \( Y \) might have several values, weighted by the function's formula. In mathematical form, \( f(y) \) tells us about the "density" of probabilities accumulated at any given point \( y \).Understanding random variables helps in making predictions and in performing statistical analysis, which can be very powerful in decision-making processes based on observed data.

The change of variables formula is a mathematical approach to transform the variable of a probability density function from one form to another. In our exercise, we want to understand how the random variable \( Y \) (the impurities) relates to another random variable \( U \) (the value in dollars). We begin by expressing \( Y \) (the initial variable) in terms of \( U \) (the new variable), which is derived as \( Y = 10 - 2U \). By using this relationship, we aim to find the density function of \( U \). This transformation process is essential when the desired quantity or insight involves a new variable or measurement, especially when performing integration over a probability density function. Properly changing variables allows us to understand how changing \( Y \) directly affects \( U \), and adjust our approach to probability accordingly.

The derivative plays a crucial role in probability theory when changing variables. It helps us understand how the value of one variable affects another variable, particularly when converting between different probability distributions. In our exercise, we compute \( \frac{dY}{dU} = -2 \) to determine how probabilities related to \( Y \) should be adjusted to find those related to \( U \). Essentially, this derivative acts as a "scaling factor," capturing how the probability density stretches or contracts as we switch from \( Y \) to \( U \). This changing of scales through differentiation is especially important when applying the change of variables formula for probability density functions. It helps maintain the integrity of the probability distribution during transformations, ensuring that it remains accurate and meaningful.

Transforming a density function is about modifying the function to describe a different variable, in this case, from \( Y \) to \( U \). This involves determining the function form of \( U \) from the known function of \( Y \). We utilize the formula:\[ f_U(u) = f_Y(g(u)) \cdot \left| \frac{dg(u)}{du} \right| \]This formula ensures that the properties of distributions are preserved and allows us to compute \( f_U(u) \), the density function for the dollar value \( U \). By substituting \( Y = 10 - 2U \) into the original density equation for \( Y \), we recalculate and simplify to determine the resulting density function:\[ f_U(u) = 12u^2 - 124u + 320 \quad \text{for} \quad 4.5 \leq u \leq 5 \]Transformations like this are common in probability when dealing with real-world data that need to be interpreted in different contexts. Understanding these transformations helps in crafting suitable models that reflect changes in defined variables.