Discussion
If one observation X from Bernoulli distribution with parameter \(p \in \left[\frac{1}{3}, \frac{2}{3}\right]\) is obtained, then the MLE of \(p\) is :
Answer: C
The MLE is \(\frac{X + 1}{3}\).
The likelihood of one Bernoulli observation is \(p^{X}(1 - p)^{1 - X}\). If \(X = 1\), it is \(p\), increasing in \(p\), so on \([1/3, 2/3]\) it is maximised at the right end \(p = 2/3\). If \(X = 0\), it is \(1 - p\), decreasing, so it is maximised at \(p = 1/3\).
The formula \((X + 1)/3\) gives exactly 1/3 for \(X = 0\) and 2/3 for \(X = 1\).
The unrestricted MLE X takes the values 0 and 1, which lie outside the parameter space. \((2X + 1)/3\) gives 1 when \(X = 1\), also outside, and \((X - 1)/3\) is negative or zero, never in \([1/3, 2/3]\).
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