---
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: 2026-11-12
---

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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 two trials landing success, failure, the likelihood is p times (1 minus p). The estimate is the parameter value that makes this function largest, and like any estimator you can judge it by bias and variance.

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 7 successes in 20 trials, the log-likelihood is 7 log p plus 13 log (1 minus p), and solving gives p equal to 7 over 20, which is 0.35. For exponential lifetimes with sample mean 5 years, the maximiser is 1 over the mean, which is 0.2 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

- [Point Estimation: Bias, Variance and Consistency](https://lightmysky.com/learn/mathematics/point-estimation-bias-variance-and-consistency-mt_I374r2UU_i)

## Optional connections

- [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT)

## Opens up

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