Studying cell division in some bacteria species

  Assume that we are studying cell division in some bacteria species. A new-born cell is divided after some time into two new cells, and the time from birth to division varies for different individual cells. We suspect that the probability distribution of the bacteria lifetimes is exponential, that is, P(t)=1τe−t/τ with some positive but unknown parameter τ. It is easy to see that the above function is normalized to 1.0 for any such τ. We have traced three bacteria and found that their lifetimes (in hours) are 24.2, 32.1 and 28.7. Use the maximum likelihood method to estimate τ. You can solve the problem graphically. Plot the likelihood function as a function of τ and determine approximately where the maximum is. One figure after a decimal point is enough. You can zoom in your plot around the maximum to increase precision as needed. Note that τ is also measured in hours. Extra Credit (5pt). This problem has an analytical solution not only for three measurements as above but for an arbitrary number of measurements ti,i=1,2,...,n. Derive the analytical solution for an arbitrary number of measurements.