For a radioactive ensemble, the number of undecayed nuclei satisfies $$\frac{dN}{dt}=-\lambda N$$ with solution $$N(t)=N_0e^{-\lambda t}$$. The half-life is $t_{1/2}=\frac{\ln 2}{\lambda}$ and the mean lifetime is $\tau=\frac{1}{\lambda}$.

Alpha decay is a quantum tunneling process through the Coulomb barrier. Beta decay is governed by the weak interaction and changes $Z$ by one unit while conserving baryon number. Gamma decay de-excites nuclei between excited states without changing $A$ or $Z$.

The valley of stability arises from the competition between symmetry energy and Coulomb repulsion. The proton and neutron drip lines define the limits beyond which adding one more nucleon leads to immediate unbinding. Near the drip lines, weak binding, continuum coupling, and halo phenomena become prominent.