---
title: "Maximum Likelihood Estimation"
description: "Write the probability of the observed data as a function of the unknown parameter and choose the parameter that makes the data most likely. Logs turn the product into a sum you can differentiate."
canonical: https://lightmysky.com/learn/mathematics/maximum-likelihood-estimation-mt_ZRTrcy7-KV
source: https://lightmysky.com/learn/mathematics/maximum-likelihood-estimation-mt_ZRTrcy7-KV.md
retrieved: 2026-09-12
---

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# Maximum Likelihood Estimation

Write the probability of the observed data as a function of the unknown parameter and choose the parameter that makes the data most likely. Logs turn the product into a sum you can differentiate.

Subject: Mathematics · Area: Data & Statistics · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/maximum-likelihood-estimation-mt_ZRTrcy7-KV

## Ready when they can

- Write a likelihood for a sample from a stated model
- Maximise the log-likelihood and verify it is a maximum
- Derive the estimator for a binomial proportion or an exponential rate

## Lesson: Making the data most likely

Write the chance of the data you actually saw as a function of the unknown parameter. That function is the likelihood. For three trials landing success, success, failure, the likelihood is p squared times (1 minus p). The estimate is the parameter value that makes this function largest.

Products are hard to differentiate, so take the log first: it turns the product into a sum and keeps the same maximiser. Differentiate the log-likelihood, set the slope to zero, and solve. Then confirm a maximum with negative curvature, since zero slope alone only marks a stationary point.

**Example.** With 12 successes in 30 trials, the log-likelihood is 12 log p plus 18 log (1 minus p), and solving gives p equal to 12 over 30, which is 0.4. For exponential lifetimes with sample mean 4 years, the maximiser is 1 over the mean, which is 0.25 per year. In each case the data's own pattern names the estimate.

**Recap.** The likelihood scores each parameter by how probable it makes the data, and the log turns maximising it into routine calculus.

## Practice

12 questions on this page, each with its working shown.

## Needs first

- [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT)
- [Point Estimation: Bias, Variance and Consistency](https://lightmysky.com/learn/mathematics/point-estimation-bias-variance-and-consistency-mt_I374r2UU_i)

## Opens up

- [Confidence Intervals for Means and Proportions](https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn)