1.

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» (a) The probability of survival(i.e., not decaying) in time `t` is `e^(-lambda t)`. Hence the probability of decay is `1-e^(- lambdat)`
(b) The probaility that the particle decays in time `dt` around time `t` is the difference
`e^(-lambda t)-e^(-lambda(t+dt))= e^(-lambdat)[1-e^(e lambda dt)]= lambda e^(-lambda t)dt`
Therefore the mean life time is
`T= int_(0)^(oo)t lambdae^(-lambdar)dt// int_(0)^(oo) lambdae^(-lambdat)dt=(1)/(lambda) int_(0)^(oo)xe^(-x)dx//int_(0)^(oo)e^(-x)dx=(1)/(lambda)`


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