${\displaystyle {\frac {dN}{dt))=-\lambda {N))$

${\displaystyle N(t)=N_{0}e^{-\lambda {t))}$

${\displaystyle \tau ={\frac {1}{\lambda ))}$

${\displaystyle N(t)=N_{0}e^{-t/\tau ))$

${\displaystyle t_{1/2}={\frac {\ln 2}{\lambda ))=\ln 2\cdot \tau .}$

${\displaystyle \tau ={\frac {t_{1/2)){\ln 2))\simeq 1.443\cdot {t_{1/2))}$

${\displaystyle N(t)=N_{0}2^{-t/t_{1/2))}$

${\displaystyle {\frac {dN}{dt))=-\lambda {N))$

${\displaystyle {\frac {dN}{N))=-\lambda {dt))$

${\displaystyle \ln N=-\lambda {t}+C}$

${\displaystyle N(t)=e^{C}e^{-\lambda {t))=N_{0}e^{-\lambda {t))}$

${\displaystyle N=N_{0}e^{-\lambda {t))}$

${\displaystyle 1=\int _{0}^{\infty }c\cdot {N_{0))e^{-\lambda {t))dt=c\cdot {\frac {N_{0)){\lambda ))}$

${\displaystyle c={\frac {\lambda }{N_{0))))$

${\displaystyle \tau =\langle t\rangle =\int _{0}^{\infty }t\cdot {c}\cdot {N_{0))e^{-\lambda {t))dt=\int _{0}^{\infty }\lambda {t}e^{-\lambda {t))dt={\frac {1}{\lambda )).}$

${\displaystyle -{\frac {dN(t)}{dt))=N\lambda _{1}+N\lambda _{2}=(\lambda _{1}+\lambda _{2})N}$

${\displaystyle N(t)=e^{-(\lambda _{1}+\lambda _{2})}=e^{-\lambda _{c))}$

${\displaystyle {\frac {1}{\tau _{c))}=\lambda _{c}=\lambda _{1}+\lambda _{2}={\frac {1}{\tau _{1))}+{\frac {1}{\tau _{2))))$

${\displaystyle \tau _{c}={\frac {\tau _{1}\tau _{2)){\tau _{1}+\tau _{2))))$

${\displaystyle T_{1/2}={\frac {t_{1}t_{2)){t_{1}+t_{2))))$

${\displaystyle T_{1/2}={\frac {\ln 2}{\lambda _{c))}={\frac {\ln 2}{\lambda _{1}+\lambda _{2))))$

${\displaystyle T_{1/2}={\frac {\ln 2}{\lambda _{c))}={\frac {\ln 2}{\lambda _{1}+\lambda _{2}+\lambda _{3))}={\frac {t_{1}t_{2}t_{3)){(t_{1}t_{2})+(t_{1}t_{3})+(t_{2}t_{3})))}$