gen-define-maximum-likelihood-estimation-and-give-one-short-work-5529

Define maximum likelihood estimation and give one short worked example.

# Maximum Likelihood Estimation (MLE) **Definition:** Maximum likelihood estimation is a method for estimating the parameters of a statistical model by finding the parameter values that maximize the likelihood function—that is, the values that make the observed data most probable. ## Worked Example **Problem:** We flip a coin 10 times and get 7 heads. Estimate the probability of heads, θ. **Solution:** 1. **Likelihood function:** For n flips with k heads following a binomial distribution: L(θ | data) = θ^k (1-θ)^(n-k) = θ^7(1-θ)^3 2. **Maximize:** Take the derivative with respect to θ and set to zero: dL/dθ = 7θ^6(1-θ)^3 - 3θ^7(1-θ)^2 = 0 Factor: θ^6(1-θ)^2[7(1-θ) - 3θ] = 0 7 - 7θ - 3θ = 0 7 = 10θ 3. **MLE:** θ̂ = 7/10 = 0.7 **Interpretation:** The maximum likelihood estimate for the probability of heads is simply the observed proportion: 7/10. This intuitively makes sense—the parameter value that makes our observed data most likely is the one that matches what we actually observed.