A probability density function (PDF) is a mathematical representation that describes the probability distribution of a continuous random variable. It is defined as a function that assigns probabilities to different values within a given range of the random variable. Specifically, it quantifies the likelihood of a random variable taking on a particular value or falling within a specific interval.
The PDF provides a continuous probability distribution, as opposed to a discrete probability distribution, which is described by a probability mass function (PMF). The PDF is commonly denoted as f(x), where x represents the random variable.
The integral of the PDF over a particular interval gives the probability that the random variable will fall within that interval. As the integral over the entire range of the random variable is equal to 1, the PDF satisfies the normalization condition. The PDF can have different shapes, including bell-shaped (e.g., normal distribution), skewed (e.g., exponential distribution), and uniform (e.g., uniform distribution). The shape of the PDF determines the characteristics of the random variable, such as the mean, variance, and skewness.
The PDF plays a crucial role in statistical analysis, hypothesis testing, and inference. It provides a framework for calculating probabilities and making predictions about continuous random variables. With the PDF, statistical models can be constructed, and various properties of the random variable can be determined, enabling the understanding and utilization of probability distributions in diverse fields such as engineering, economics, and social sciences.
