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To determine whether a mathematical function behaves as a probability density function (PDF), we must confirm several properties. This involves verifying if the function is appropriately defined over the specified domain. For instance, the function given is \(f(x)=\frac{8}{9} x\), valid within the bounds \[0 \leq x \leq \frac{3}{2}\]. Begin by checking if this expression is non-negative throughout the stated range. This ensures the output of the function does not include any negative probabilities, which do not exist in probability theory. This step helps confirm the legitimacy of the function before moving forward. Doing such verification is essential, as it sets a foundation for ensuring the function adheres to the rules of probability.

The next crucial step is calculating the integral of the function over the given interval. This process involves computing the definite integral within the prescribed limits. For example, to validate \(f(x)=\frac{8}{9} x\), we determine the integral from 0 to \frac{3}{2}\: \(\begin{aligned} \int_{0}^{3/2} \frac{8}{9} x \ dx= \left. \frac{8}{9} \cdot \frac{x^{2}}{2} \right|_{0}^{3/2} = \frac{8}{9} \cdot \frac{(3/2)^{2}}{2} - 0 = \frac{8}{9} \cdot \frac{9}{8} = 1. \end{aligned}\). A valid PDF must have an integral equal to 1 over its domain, which signifies that the total probability of all outcomes sums up to 1. Computing integrals allows one to consolidate the function's attributes with respect to its area under the curve.

In verifying a PDF, one must ensure the function's non-negativity over the specified domain. A negative value for a probability function would be nonsensical, as probabilities range from 0 to 1. For the function \(f(x)=\frac{8}{9} x\), examine the interval from 0 to \frac{3}{2}. By simply observing, \(f(x)\) remains non-negative throughout because it is a product of two positive terms within the given range. This non-negativity check solidifies that \(f(x)\) does not output any invalid negative probabilities, making it a foundational aspect of any PDF's verification.

Probability theory forms the bedrock of understanding PDFs. A PDF describes how probabilities are distributed over outcomes, meaning it must adhere strictly to the rules of probability. For a function to be a PDF:

• The function must be non-negative over its entire domain.
• The total area under the function from the start to the end of the domain must equal 1.

In this exercise, the function \(f(x)=\frac{8}{9} x\) meets these criteria over the given interval \[0 \leq x \leq \frac{3}{2}\]. Thus, intuition from probability theory confirms that \(f(x)\) effectively measures the spread of probabilities across its domain. Comprehending these fundamental constructs allows one to assess and validate functions in the realm of probability.