Describe the concepts of ‘hazard function’ and ‘cumulative hazard function’.

The hazard function and cumulative hazard function are important concepts in survival analysis, which is a branch of statistics that deals with the study of time-to-event data, such as the time until death, failure, or some other event of interest. These concepts are used to model the risk or probability of an event occurring at a certain point in time, given that the event has not occurred up to that point.

The hazard function, also known as the instantaneous hazard rate or force of mortality, is a measure of the risk of an event occurring at a given time, given that the event has not yet occurred. The hazard function is defined as the ratio of the probability of an event occurring in a small time interval to the probability of survival up to that time interval. Mathematically, the hazard function _____ ____ _____ ___ _________ _________ ____ ___ ___ ______ ____.
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