4. Maximum likelihood meets histograms

Let X1, X2, · · · , Xn be n i.i.d data points drawn from a piece-wise constant probability density function over N equal size bins between 0 and 1 (B1, B2, · · · , BN ), where the constants are θ1, θ2, · · · , θN .

(a) Using the fact that the total area underneath a probability density function is 1, express θN in terms of the other constants.

(b) Write down the log-likelihood of the data in terms of θ1, θ2, · · · , θN−1 and µ1, µ2, · · · , µN−1.

(c) Find the maximum likelihood estimate of θj for j ∈ {1, 2, · · · , N}.