Radioactive Decay — Question 2

Question 2

Mean lifetime differs from the median decay time

A single nucleus has lifetime probability density $f(t)=\lambda e^{-\lambda t}$ for $t\ge0$.

1. Calculate its mean lifetime and variance.

2. For half-life 8.00 h, find mean lifetime and the conditional probability of surviving another 8.00 h after already surviving 40.0 h.

Question 2 - Solution

Mean lifetime differs from the median decay time

(a) Method and calculation

$$\langle t\rangle=\int_0^\infty t\lambda e^{-\lambda t}dt=\boxed{1/\lambda}.$$$$\langle t^2\rangle=2/\lambda^2\Rightarrow\boxed{\operatorname{Var}(t)=1/\lambda^2}.$$

(b) Interpretation and further calculation

$$\boxed{\tau=T_{1/2}/\ln2=11.54\,\mathrm h}.$$$$\boxed{P(t>48\mid t>40)=e^{-\lambda48}/e^{-\lambda40}=e^{-\lambda8}=1/2}.$$

Physical check

Exponential decay is memoryless. Surviving for several half-lives does not make a particular nucleus more likely to decay during the next fixed interval.