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The Probability Density Function (PDF) is a fundamental concept in continuous probability distributions. For a continuous random variable, the PDF, represented as \( f(x) \), describes the likelihood of the random variable taking on a particular value. In a graphical sense, it's the function used to specify the probability of the variable within a certain range of values.

Key characteristics of a PDF include:

• Non-negativity: \( f(x) \geq 0 \) for all \( x \), ensuring probabilities aren't negative.
• Normalization: The total area under the curve of the PDF across its entire range equals one, i.e., \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \). This property ensures that the sum of all possible probabilities is 1.

In the Rayleigh distribution, the PDF is \( f(x; \theta) = \frac{x}{\theta^2} \cdot e^{-x^2 / (2\theta^2)} \) for \( x > 0 \). This formula models the vibratory stress on a wind turbine blade in our given problem. The parameter \( \theta \) is known as the scale parameter, influencing the shape of the distribution curve.

To verify a legitimate PDF, the conditions above must be checked, as demonstrated in the solution by ensuring the PDF integrates to 1 over its defined range.

The Cumulative Distribution Function (CDF) is an integral part of understanding continuous probability distributions. It provides the probability that a random variable takes on a value less than or equal to a specific point. In essence, the CDF is the sum of probabilities up to a maximum value.

For the Rayleigh distribution, the CDF is given by:

\[ F(x) = 1 - e^{-x^2/(2\theta^2)} \]

This function smoothly captures the cumulative probability from the lower bound to any point \( x \).

• This equation allows us to calculate probabilities up to any value of \( x \).
• For example, \( P(X \leq 200) \) results in 0.8647, representing an 86.47% chance that stress values are 200 psi or less.

When working with continuous distributions like the Rayleigh, \( P(X \leq x) \) and \( P(X < x) \) are effectively the same due to the continuous nature, eliminating the gap between less than and equal to.

The CDF's insight deepens our understanding of distribution behavior beyond the PDF's specifics, capturing the full range of probabilities in a single function.

Continuous Probability Distributions deal with random variables that can take on an infinite number of values within a given range. Unlike discrete distributions with countable outcomes, continuous distributions allow for any value along an interval on the number line.

Key aspects of continuous distributions include:

• Probability Density Function (PDF): Used to define the relative likelihood of a variable within a range of values.
• Continuous nature: No gaps exist between possible outcomes, as the variable can assume any value in a specified continuum.

In our context, the Rayleigh distribution is a prime example of a continuous probability distribution, often used in scenarios where phenomena are influenced by random fluctuations around a known average. The calculation of probabilities in continuous distributions involves integrating the PDF over the variable's desired range to determine likelihoods.

For instance, when determining probabilities like \( P(100 \leq X \leq 200) \) for stress values, integration of the PDF allows us to pinpoint the chance that the random variable \( X \) falls within a specified interval, leveraging the continuity of possible outcomes.