Estimating Coin's Head Probability via Maximum Likelihood

Solution & Explanation

The problem involves estimating the probability of a biased coin landing on heads using Maximum Likelihood Estimation (MLE). Here's a breakdown of the solution:

Step 1: Understanding the Problem

• N: Total number of coin flips.
• K: Number of times the coin landed on heads.
• p: Probability of the coin landing on heads (unknown).

Step 2: Setting Up the Likelihood Function

The likelihood function for observing K heads in N flips, assuming each flip is independent and identically distributed, follows a binomial distribution:

$L(p | K, N) = \binom{N}{K} p^K (1-p)^{N-K}$

Where:

• $\binom{N}{K}$ is the binomial coefficient, which is constant with respect to p.
• $p^K$: Probability of observing K heads.
• $(1-p)^{N-K}$: Probability of observing N-K tails.

Step 3: Converting to Log-Likelihood

To simplify differentiation, we take the natural logarithm of the likelihood function:

$\log L(p) = \log \binom{N}{K} + K \log(p) + (N-K) \log(1-p)$

The binomial coefficient term is constant concerning p, so it can be ignored for MLE purposes.

Step 4: Differentiating the Log-Likelihood

Differentiate the log-likelihood function with respect to p:

$\frac{d}{dp} \log L(p) = \frac{K}{p} - \frac{N-K}{1-p}$

Step 5: Setting the Derivative to Zero

To find the maximum likelihood estimate, set the derivative equal to zero:

$\frac{K}{p} - \frac{N-K}{1-p} = 0$

Step 6: Solving for p

Rearrange and solve the equation:

$\frac{K}{p} = \frac{N-K}{1-p}$

$K(1-p) = p(N-K)$

$K - Kp = pN - pK$

$K = pN$

$p = \frac{K}{N}$

This result makes intuitive sense: the probability of heads is estimated by the proportion of heads observed in the total number of flips.

Conclusion

The Maximum Likelihood Estimator (MLE) for the probability of a biased coin landing on heads is simply the ratio of the number of heads observed to the total number of flips, $p = \frac{K}{N}$. This estimator maximizes the likelihood of observing the given data under the assumption of a binomial distribution.