Maximum Likelihood Estimation for Loan Default Risk

Requirements Clarification & Assessment

1. Objective:

   • Develop a model to predict the probability $p$ of loan default using historical data.

2. Data Provided:

   • Historical outcomes of ten loans recorded as binary values: 0 for "no default" and 1 for "default".
   • Data: [0, 1, 0, 0, 1, 0, 0, 1, 0, 0].

3. Assumptions:

   • Distribution: Assume the loan outcomes are independent and identically distributed (i.i.d.) Bernoulli trials, as each loan can either default or not.
   • Parameter: $p$ represents the probability of default for each loan.

4. Tasks:

   • Derive the log-likelihood function for $p$.
   • Compute the Maximum Likelihood Estimate (MLE) for $p$.

5. Constraints:

   • The solution should be based solely on the provided data without external datasets or assumptions beyond those stated.

6. Performance Metrics:

   • Correctness of the MLE calculation.
   • Understanding of the log-likelihood function.
   • Ability to interpret results in the context of loan default risk.