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Question

(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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Solution

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(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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