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Survival analysis is a statistical approach that uses the survival function, S(t), to predict the time until an event occurs, such as death or system failure. This function, which declines from 1 to 0 over time, is complemented by the hazard and cumulative hazard functions, providing insights into the immediate and cumulative risks of the event. The exponential survival function is a special case with constant hazard rates, useful in certain predictive models. Understanding these functions is crucial for applications in healthcare, engineering, and finance.

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## Survival Function

### Definition

The survival function, denoted as S(t), is a key tool used to estimate the likelihood of survival past a specified time without experiencing a particular event

### Construction

Time-to-event data

The survival function is constructed from time-to-event data, which captures the elapsed time before an event occurs

Probability curve

The survival function is a probability curve that decreases from 1 to 0 over time, reflecting the increasing probability of the event with the passage of time

### Practical implications

The survival function has practical implications in healthcare, engineering, and finance, among other domains, for predicting the timing and probability of events

## Hazard Function

### Definition

The hazard function, denoted as h(t), quantifies the immediate risk of an event occurring at time t, given that it has not occurred yet

### Cumulative Hazard Function

The cumulative hazard function, denoted as H(t), aggregates the hazard over time and provides a cumulative measure of risk

### Relationship with Survival Function

The hazard function is the negative derivative of the natural logarithm of the survival function, providing insights into the risk profile of an event over time

## Exponential Survival Function

### Definition

The exponential survival function, represented as S(t) = e^(-λt), assumes a constant hazard rate over time and is useful for predicting survival times in situations where the event's likelihood is not time-dependent

### Application

The exponential survival function is commonly used in predicting the lifespan of machinery and chemical processes

## Baseline Survival Function

### Definition

The baseline survival function, denoted as S0(t), represents the survival probabilities under a standard set of conditions and serves as a benchmark for comparing the impact of various factors on survival times

### Importance

The baseline survival function is essential for understanding the effects of different factors on survival times and making informed decisions in areas such as clinical trials and reliability testing