Question

A biased coin, with probability of head as \( p \), is tossed \( m \) times independently. It is known that \( p \in \left\{ \frac{1}{4}, \frac{3}{4} \right\} \) and \( m \in \{3, 5\}. \) If 3 heads are observed in these \( m \) tosses, then which of the following statements is correct?

• A. \((3, \frac{3}{4})\) is a maximum likelihood estimator of \((m, p)\)
• B. \((5, \frac{1}{4})\) is a maximum likelihood estimator of \((m, p)\)
• C. \((5, \frac{3}{4})\) is a maximum likelihood estimator of \((m, p)\)
• D. Maximum likelihood estimator of \((m, p)\) is NOT unique

Answer 1

The Correct Option is A

1. Likelihood Function: The probability of observing 3 heads is given by the binomial probability: \[ L(m, p) = \binom{m}{3} p^3 (1-p)^{m-3}. \] 2. Maximum Likelihood Estimation: - Evaluate \( L(m, p) \) for each pair \( (m, p) \) from the possible values: \[ L(3, \frac{3}{4}) > L(5, \frac{1}{4}), \quad L(3, \frac{3}{4}) > L(5, \frac{3}{4}). \] - The pair \( (3, \frac{3}{4}) \) maximizes the likelihood function