Half-Life¶

Definition¶

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

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 $h$ with probability $\tfrac{1}{2}$ , so $h$ is a median of $X$ . 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 $S_X(t)=\mathrm{e}^{-t}$ $e^{- t}$ , an exponential lifetime with rate $\lambda=1$ $1$ . It reaches $\tfrac{1}{2}$ at $h=\ln 2\approx0.69$ $0.69$ , then $\tfrac{1}{4}$ at $2h$ and $\tfrac{1}{8}$ at $3h$ : equal steps in time halve the survival probability each time. In formulas, for integers $k\ge0$ $0$ ,

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

Change the rate $\lambda$ : the half-life $h=\ln 2/\lambda$ is where survival drops to $\frac{1}{2}$ , and each further $h$ halves it again.

Solving $\mathrm{e}^{-\lambda h}=\tfrac{1}{2}$ with the natural logarithm gives $h=\ln 2/\lambda$ $ln 2/ λ$ . It is shorter than the mean lifetime $1/\lambda$ by the factor $\ln 2\approx0.69$ $0.69$ : 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 $h$ , half also last a further $h$ . For a lifetime that is not exponential, the Definition still makes sense, but the halving pattern fails: the time from $\tfrac{1}{2}$ to $\tfrac{1}{4}$ need not equal $h$ . If $S_X$ stays at $\tfrac{1}{2}$ 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 $\lambda$ , the half-life is $\ln 2/\lambda$ .
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¶

Arrival Rate

Sources¶

Probability Theory for EOR · Week 6

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

Last updated: 20 October 2026 03:02