The Probability Density Function, or PDF, is a fundamental concept in probability theory and statistics. For the gamma distribution, the PDF determines how probability is distributed over different values. It is expressed as a function. The PDF of the gamma distribution is given by \(f(x) = \frac{1}{\beta^2} x e^{-x/\beta}\) for \(0 < x < \infty\) and is zero elsewhere. Here, \(x\) represents a possible outcome, and \(\beta\) is a positive parameter that affects the shape of the distribution.
- \(x\) is the random variable associated with the gamma distribution.
- \(\beta\) is known as the scale parameter, controlling how "spread out" the distribution is.
- The exponent \(e^{-x/\beta}\) brings the exponential nature to the distribution, making it tail off as \(x\) increases.
To work with the gamma PDF, you first need to determine all necessary parameters, such as \(\beta\). Without the correct parameters, the PDF won't accurately represent the actual probability distribution.