Let f denote the density function and h the hazard function...

Let $f$ denote the density function and $h$ the hazard function of a nonnegative random variable. Show that
$$
f(t)=h(t) e^{-\int_{0}^{t} h(s) d s}
$$
that is, that the hazard function uniquely determines the density.

Key Concepts

Density Function

The density function, often denoted by f(t), represents the rate at which probability is accumulating at time t for a continuous random variable. It is the derivative of the cumulative distribution function (CDF) and satisfies the property that the integral over its entire domain equals 1. This function provides a complete description of the underlying probability distribution when it exists.

Hazard Function

The hazard function, denoted by h(t), is a measure used in survival analysis and reliability theory which quantifies the instantaneous risk of an event occurring at time t, given that the event has not occurred before t. It represents the conditional probability rate and offers insight into how the risk evolves over time.

Survival Function

The survival function, typically expressed as S(t) = 1 - F(t) where F(t) is the cumulative distribution function, gives the probability that the event of interest has not occurred by time t. It is a foundational concept in survival and reliability studies and provides an alternative perspective on the distribution of lifetimes or durations.

Relationship Between Hazard and Density Functions

The key relationship between the hazard function and the density function is established through the survival function. Specifically, the survival function can be expressed as S(t) = exp(-??? h(s) ds), where the integral of the hazard function represents the cumulative risk over time. Differentiating this expression with respect to time leads to the density function expressed as f(t) = h(t) exp(-??? h(s) ds), thereby uniquely determining the density function from the hazard function.

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