The hazard function is a statistical tool used in survival analysis to describe the instantaneous risk of an event occurring at a specific time, given that the event has not yet happened. It helps to understand the likelihood of failure or death over time, making it useful in fields like medicine and engineering. Mathematically, the hazard function is defined as the ratio of the probability density function to the survival function. It provides insights into the timing of events, allowing researchers to model and predict outcomes related to patients, machines, or any subject under study.