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Half-Life

Definition

Let XX be a nonnegative random variable with survival function SX(t)=P(X>t)S_X(t)=\mathbb{P}(X>t), also written F‾X(t)\overline F_X(t). A half-life is a time hh such that

SX(h)=12.S_X(h)=\frac{1}{2}.

Interpretation

The half-life is the time by which half of a population of lifetimes has ended: a lifetime exceeds hh with probability 12\tfrac{1}{2}, so hh is a median of XX. For a decaying quantity it is the time after which half of the initial amount is left.

The survival curve e to the minus t with the points where it reaches 1/2, 1/4 and 1/8, at one, two and three half-lives.

The curve is SX(t)=e−tS_X(t)=\mathrm{e}^{-t}, an exponential lifetime with rate λ=1\lambda=1. It reaches 12\tfrac{1}{2} at h=ln⁡2≈0.69h=\ln 2\approx0.69, then 14\tfrac{1}{4} at 2h2h and 18\tfrac{1}{8} at 3h3h: equal steps in time halve the survival probability each time. In formulas, for integers k≥0k\ge0,

SX(kh)=(e−λh)k=2−k.S_X(kh)=\big(\mathrm{e}^{-\lambda h}\big)^k=2^{-k}.

Solving e−λh=12\mathrm{e}^{-\lambda h}=\tfrac{1}{2} with the natural logarithm gives h=ln⁡2/λh=\ln 2/\lambda. It is shorter than the mean lifetime 1/λ1/\lambda by the factor ln⁡2≈0.69\ln 2\approx0.69: a few very long lifetimes pull the mean up, while the half-life, being a median, does not depend on how long the longest lifetimes last.

The repeated halving is the memoryless property at work: of the lifetimes that last beyond hh, half also last a further hh. For a lifetime that is not exponential, the Definition still makes sense, but the halving pattern fails: the time from 12\tfrac{1}{2} to 14\tfrac{1}{4} need not equal hh. If SXS_X stays at 12\tfrac{1}{2} on a whole interval, every time in it solves the equation, which is why the Definition speaks of a half-life.

Key Results

  • For exponential decay with rate λ\lambda, the half-life is ln⁡2/λ\ln 2/\lambda.
  • Half-life is a median survival time in exponential waiting models.
  • Shorter half-life corresponds to faster decay or faster event occurrence.

Examples

  • If a drug concentration decays exponentially, the half-life is the time until the concentration is halved.
  • In radioactive decay, half-life summarizes how quickly unstable atoms decay.
  • For an exponential component lifetime, the half-life is the time by which half of components are expected to have failed.

Prerequisites

Sources

Keywords: median lifetime, exponential half-life Symbols: lambda

Last updated: 20 October 2026 03:02