Set k=θ/(1−θ),so k≥1 and the density is kx^(k−1). The unconstrained likelihood maximizer is k̂=−4/Σln x_i. Here Σln x_i≈−5.4553,so k̂≈.733<1. On the permitted range the maximum is at k=1,corresponding to θ=1/2. Answer **0.50**.
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