Probablity distribution for two particles to decay?

Question

Let us say I have the probability distribution of the decay of one particle as: $$f(t)=\frac{1}{\tau}e^{-\frac{t}{\tau}}$$ Then how would I find the probablity distribution for the time it takes two (specific) particles to decay (given that they have the same probablity distribution as given above, and that the decay of one particle is independent of that of the other)?

Answer 1

If $T_i$ denotes the decay of particle $i$ then we can find the CDF of $M:=\max(T_1,T_2)$ by: $$P(M\leq t)=P(T_1\leq t\wedge T_2\leq t)=P(T_1\leq t)P(T_2\leq t)=(1-e^{\frac{-t}{\tau}})^2$$

The PDF of $M$ can be found by differentiating the CDF.