# Reliability (survival) function R(t)

Operations · `Reliability or survival function` · verification: source checked

The reliability or survival function R(t) gives the probability that a unit survives beyond a given time t. Because a unit either fails or survives, R(t) = 1 − F(t), where F(t) is the probability of failure by time t. For a system of independent components, the system reliability is the product of the reliability functions of all the components, since every component must survive for the system to survive.

## Specification

| Property | Value |
|---|---|
| Kind | metric |

## Formula

```
R(t) = 1 - F(t)
```

- `R(t)`: Reliability (survival) function: probability a unit survives beyond time t (probability)
- `F(t)`: Failure probability by time t (probability)

## Study notes

- Survival is the complementary event to failure; one of the two mutually exclusive alternatives must occur.
- F(t) = 1 − R(t), so the failure distribution and the reliability function carry the same information.
- The reliability of a system of independent components is the product of the reliability functions of all the components.
- The product rule holds because every component must survive in order for the system to survive.
- The document states that building the system reliability up from individual components is treated in detail under the "Bottom-Up" method.

## Sources

1. [8.1.2.2. Reliability or survival function](https://www.itl.nist.gov/div898/handbook/apr/section1/apr122.htm) (accessed 2026-10-25)

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