Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 6 / iv / Solution

For these counts, start with any $0<{\theta }^{(0)}<1$. The conditional missing count from part (ii) has meanThe E-step objective isIts strictly concave M-step is explicit:Equivalently, the update map is $F(\theta )=(35+360\theta )/(245+480\theta )$. Its fixed-point equation isThe unique root in $(0,1)$ is the maximum-likelihood estimateFor example, starting at $1/2$ gives $0.443299$, then approximately $0.425065$, and subsequent iterates approach the displayed root. Convergence is also transparent without relying only on the general EM theorem: $F$ is increasing andThis is positive below the feasible fixed point and negative above it. Monotonicity of $F$ prevents crossing that fixed point, so the iterates converge monotonically toward it from either side. The original log-likelihood is strictly concave and diverges to minus infinity at both endpoints, establishing that the fixed point is the unique global maximum.