Question

Knowing the decay constant $$ \lambda $$ of a nucleus, find
(a) the probability of decay of the nucleus during the time from $$0$$ to $$t$$;
(b) the mean lifetime $$\tau $$ of the nucleus.

Answer 1

(a) The probability of survival (i.e., not decaying) in time $$t$$ is $$e^{-\lambda t} $$. Hence the probability of decay is $$ 1 - e^{-\lambda t} $$.
(b) The probability that the particle decays in time $$dt$$ around time $$t$$ is the difference
$$e^{-\lambda t} - e^{-\lambda(t + dt)} = e^{-\lambda t} [ 1 - e^{-e \lambda\ dt}] = \lambda e^{-\lambda t} dt $$
Therefore the mean life time is
$$ \displaystyle T = \int^{\infty}_{0} t \lambda e^{-\lambda t} dt / \int^{\infty}_{0} \lambda e^{-\lambda t} dt = \dfrac{1}{\lambda} \int^{\infty}_{0} xe^{-x} dx / \int^{\infty}_{0} e^{-x} dx = \dfrac{1}{\lambda} $$