The steps are the same as in the case of photon survival.
On average, how much time will pass before a radioactive atom decays?
Again, we find a "chance" process being described by an exponential decay law.
We can easily find an expression for the chance that a radioactive atom will "survive" (be an original element atom) to at least a time t.
However, now the "thin slice" is an interval of time, and the dependent variable is the number of radioactive atoms present, N(t). If we have a sample of atoms, and we consider a time interval short enough that the population of atoms hasn't changed significantly through decay, then the proportion of atoms decaying in our short time interval will be proportional to the length of the interval.
Note that that the domain of F is the interval from zero to 1, which corresponds to the interval of time from zero to infinity.
Plotting t against F with a value of l=1 gives the graph on the right. The equivalent thickness for the medium in radiation attenuation is known as "half-value thickness".
This question can be answered using a little bit of calculus. Once we have an expression for t, a "definite integral" will give us the mean value of t (this is how "mean value" is defined).
From the equation above, taking logarithms of both sides we see that lt = -ln(N/N.