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# 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.

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#### (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 $$\displaystyle T = \int^{\infty}_{0} t \lambda e^{-\lambda t} dt / \int^{\infty}_{0} \lambda e^{-\lambda t} dt = \dfrac{1}{\lambda} \int^{\infty}_{0} xe^{-x} dx / \int^{\infty}_{0} e^{-x} dx = \dfrac{1}{\lambda}$$

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