1. Definition
Maximum Likelihood Estimation (MLE) picks θ maximizing the likelihood L(θ)=∏f(xᵢ;θ). For normal: μ̂=sample mean, σ̂²=variance ÷n (biased).
2. Symbols
| Symbol | Meaning |
|---|
| L(θ) | likelihood function |
| μ̂ | MLE of normal mean |
| σ̂² | MLE of normal variance (÷n) |
| p̂ | MLE of Bernoulli p |
3. How it works
- Write the likelihood and take logs;
- Differentiate and set to zero;
- Normal: μ̂=mean, σ̂²=Σ(xᵢ−x̄)²/n;
- Bernoulli: p̂=success proportion.
4. Steps
- Choose a distribution;
- Enter sample data;
- Click Calculate;
- Distinguish MLE (÷n) from unbiased (÷n−1).
5. Example
Example: normal data 2,4,4,6,6,6,8,8,10. μ̂=6, σ̂²=48/9≈5.333.
6. Pitfalls
MLE variance divides by n, unlike unbiased sample variance (÷n−1);
MLE is consistent but not always unbiased;
Bernoulli data should be 0/1.