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The Rayleigh distribution is a continuous probability distribution, which is often used in applications involving random vectors with magnitude and direction. It's particularly important in fields like mechanical engineering and telecommunications.
This distribution is defined by the probability density function (pdf) given by \( f(x; \theta) = \frac{x}{\theta} e^{-x^2 / (2\theta)} \)
where \( x > 0 \) and \( \theta \) is a parameter known as the scale parameter. The Rayleigh distribution is typically used to model the magnitude of a vector that has normally distributed components, which makes it useful for problem contexts such as wind speeds, signal integrity, or mechanical stress vibrations.

The probability density function (pdf) is a key concept when working with continuous probability distributions such as the Rayleigh distribution. It describes the likelihood of a random variable to take on a particular value. For Rayleigh, the pdf is
\[ f(x; \theta) = \frac{x}{\theta} e^{-x^2 / (2\theta)} \]
Here, \( \theta \) is the scale parameter and dictates how "spread out" the distribution is.
Understanding the pdf provides valuable insight into how values are distributed and can be used to compute probabilities over various intervals. For instance, the likelihood of the vibratory stress on a turbine blade exceeding a certain level under specific conditions can be deduced using this function.

The expected value, often called the mean, is a measure of the center of a distribution. In the context of the Rayleigh distribution, it provides us a way to determine the average outcome we might expect for squared observations.
Given for the Rayleigh distribution, the expected value of the square of the random variable \( X \) is \( E(X^2) = 2\theta \). This implies that if we take many samples from a Rayleigh distribution, the average of these squared values should approximate \( 2\theta \).
Expected value computations are intuitively useful because they help us relate sample data to theoretical insights, like those needed for estimating parameters such as \( \theta \).

A critical task in statistics is finding unbiased estimators, which are statistical measures that reflect a true parameter value without systematic error. From a sample, unbiased estimator for \( \theta \) can be derived from the Rayleigh distribution based on the sum of squared observations.The formula for the estimator in the exercise is\[ \hat{\theta} = \frac{1}{2n} \sum_{i=1}^{n} X_i^2 \]where \( n \) is the sample size. To ensure that this estimator is unbiased, we establish that its expected value also equals \( \theta \). Processing sample data using this unbiased method gives us a calculated estimate that holds true average accuracy.Using this approach with the provided observations, we find that \( \hat{\theta} = 108.95 \), which means that the estimator closely mirrors the actual parameter value of the sample's underlying distribution.