91Ó°ÊÓ

In statistical analysis, the Probability Density Function (PDF) is a fundamental concept used to define the likelihood of a continuous random variable taking on a particular value. For the gamma distribution, the PDF enables us to describe the distribution of the random variable's outcomes, characterizing its behavior over the range of possible values it could take.When dealing with a gamma distribution, the PDF is expressed as:\[ f_X(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} \]Here:

• \(\alpha\) is the shape parameter,
• \(\beta\) is the rate parameter,
• \(\Gamma(\alpha)\) is the gamma function which generalizes the factorial function.

This formula showcases how the gamma PDF spreads out the probabilities across different values of \(X\). In our particular problem, it aids in the transformation analysis by showing how probabilities are rescaled when transforming \(X\) to \(Y = 60X\).

Random variable transformation involves altering a random variable through a mathematical function to derive a new variable. It's a key concept when changing scales or units, such as converting hours to minutes for measurements in our example problem.

Transformation Process

In our example, the transformation is moving from hours \(X\) to minutes \(Y\) through the function \(Y = 60X\). This is an example of a linear transformation. The inverse transformation required is \(X = \frac{Y}{60}\).To find the PDF of the transformed variable \(Y\), we use the formula:\[ f_Y(y) = f_X(g^{-1}(y)) \cdot \left| \frac{d}{dy} g^{-1}(y) \right| \]This technique ensures that the area under the PDF remains unified as the variable is transformed.

Applying to the PDF

By substituting this into our formula, we observe how \(f_X(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}\) adjusts through the transformation, resulting in the new PDF for \(Y\) with adjusted parameters \(\alpha\) and \(\frac{\beta}{60}\).

A probability distribution describes how probabilities are assigned across the range of a random variable. It provides a complete description of the behavior and characteristics of the variable by listing all possible outcomes and their likelihood.

Gamma Distribution Characteristics

In the context of the gamma distribution used in our problem, the parameters \(\alpha\) (shape) and \(\beta\) (rate) control the form and spread of the distribution.

Impact of Transformation

When the variable \(X\) with gamma distribution parameters \(\alpha\) and \(\beta\) is transformed to \(Y = 60X\), the probability distribution of \(Y\) is also a gamma distribution. However, its rate parameter transforms to \(\frac{\beta}{60}\). This maintains the shape \(\alpha\) while adjusting the rate of decay to fit the new scale.Understanding these transformations helps illustrate how different scales affect the distribution's spread and behavior. It emphasizes the flexibility of the gamma distribution to model various real-world processes across different units of measurement.