This question involves finding the maximum likelihood estimate (MLE) of $p$, likely based on a statistical model such as the binomial or geometric distribution. Let me explain step-by-step how to approach this and solve it. sympy import symbols, Eq, solve

Sample data provided in the problem

sample = [0, 0, 1, 2, 2, 3, 3, 3, 3, 3]

Total observations and sum of the sample data

n = len(sample) sum_sample = sum(sample)

Define the variable for probability p

p = symbols('p')

The likelihood function for geometric distribution (MLE estimation) is based on P(X=x) = (1-p)^(x-1) * p

The log-likelihood simplifies to: L(p) = n * log(p) + sum(Xi-1) * log(1-p)

The derivative of the log-likelihood with respect to p and set to 0 gives the MLE:

log_likelihood = n * p + sum_sample * (1 - p) mle_equation = Eq(log_likelihood, 0)

Solve for p

mle_solution = solve(mle_equation, p) mle_solutionIt seems the calculation produced a value of 2 for $p$, which is incorrect since probabilities must be between 0 and 1. I’ll reevaluate the setup and clarify the likelihood function.

Given the sample data, it is important to verify whether we are dealing with a binomial or geometric model. Let me revise and correctly analyze this case. Correct approach: Geometric distribution log-likelihood

For a geometric distribution, P(X=x) = (1-p)^(x-1) * p

Log-likelihood is: L(p) = n * log(p) + sum(Xi-1) * log(1-p)

from sympy import log

Define log-likelihood function for geometric distribution

log_likelihood_correct = n * log(p) + sum_sample * log(1 - p)

Derivative of log-likelihood with respect to p

mle_equation_correct = log_likelihood_correct.diff(p)

Solve the equation to find the MLE of p

mle_solution_correct = solve(Eq(mle_equation_correct, 0), p) mle_solution_correctThe maximum likelihood estimate (MLE) for $p$ is $\frac{1}{3}$, or approximately $0.33$ when rounded to two decimal places.

Would you like a detailed breakdown of the calculations or any clarifications? Here are five related questions for practice:

1. How is the likelihood function constructed for a geometric distribution?
2. What steps are involved in finding the MLE for a parameter?
3. How does the log-likelihood simplify the process of finding the MLE?
4. Why must probabilities $p$ lie between 0 and 1 in this context?
5. Can this approach be generalized to other probability distributions?

Tip: Always check the domain of $p$ (e.g., $0 < p < 1$) when solving for probabilities in likelihood estimation.