A Probability Density Function (PDF) is a function that describes the likelihood of a random variable to take on a particular value. For continuous random variables, the PDF is used to specify the probability of the variable falling within a certain range of values, as opposed to taking on any one value. This range probability is found by calculating the area under the PDF’s curve over that interval.
In the given exercise, the PDF for a random variable follows the formula:
- \[f(x ; heta)=\frac{\theta \exp \left\{-|x|^{\theta}\right\}}{2 \Gamma(1 /\theta)} , -\infty < x < \infty\]
The parameter \(\theta\) influences the shape of the PDF. Here, \(\theta\) must be greater than 0, which ensures that the PDF is well-defined. Understanding this PDF is crucial for performing hypothesis testing on the parameter, as it governs the distribution of the data. The ancient Greek letter \(\Gamma\) in the expression represents the gamma function, which generalizes the factorial function for real and complex numbers. This function often appears in probability distributions, normalizing the areas to sum to unity.