Maximum Likelihood Estimator Calculator

Calculator Input

Distribution Model

Decimal Precision

Graph Bins / Bars

Confidence Level

Sample Data

Use commas, spaces, semicolons, or line breaks. Poisson needs non-negative integers. Bernoulli needs only 0 and 1 values.

Example Data Table

Distribution	Example Sample Data	Expected MLE Idea
Normal	5.2, 4.9, 6.1, 5.8, 5.0, 4.7, 5.6, 5.3	Estimate mean μ and standard deviation σ from continuous data.
Exponential	0.8, 1.6, 2.4, 0.7, 1.1, 3.0, 2.2, 1.4	Estimate rate λ using the reciprocal of the sample mean.
Poisson	3, 2, 5, 4, 1, 0, 3, 2, 4, 3	Estimate count rate λ using the sample mean of events.
Bernoulli	1, 0, 1, 1, 0, 1, 0, 1, 1, 0	Estimate success probability p from binary outcomes.

Formula Used

General Maximum Likelihood Idea

For sample values x₁, x₂, ..., xₙ, choose the parameter that maximizes the likelihood function L(θ) = ∏ f(xᵢ | θ).

In practice, the calculator maximizes the log-likelihood ℓ(θ) = Σ log f(xᵢ | θ) because it is more stable and easier to compute.

Normal Distribution

If xᵢ ~ N(μ, σ²), then the MLEs are: μ̂ = (Σxᵢ)/n and σ̂² = (Σ(xᵢ - μ̂)²)/n.

The log-likelihood at the optimum is ℓ = -(n/2)[log(2πσ̂²) + 1].

Exponential Distribution

If xᵢ ~ Exp(λ), the MLE is λ̂ = n / Σxᵢ = 1 / x̄.

The fitted expected waiting time is 1 / λ̂.

Poisson Distribution

If xᵢ ~ Poisson(λ), then the MLE is λ̂ = x̄.

The Poisson model also implies model variance Var(X) = λ̂.

Bernoulli Distribution

If each observation is either 0 or 1, then the MLE of the success probability is p̂ = (Σxᵢ)/n.

This equals the observed proportion of ones in the sample.

Model Selection Metrics

The calculator also reports AIC = 2k - 2ℓ and BIC = k ln(n) - 2ℓ, where k is the number of estimated parameters.

Smaller AIC or BIC values indicate a better trade-off between fit and complexity for the same dataset.

How to Use This Calculator

Select the distribution that matches your data generation process.
Paste the sample values into the data box using commas or new lines.
Choose the decimal precision, graph bins, and confidence level.
Click Estimate Parameters to compute the MLEs.
Review the result summary, parameter table, log-likelihood, AIC, BIC, and chart.
Use the CSV or PDF buttons to export your output for reports or coursework.

Frequently Asked Questions

1) What does maximum likelihood estimation do?

It finds the parameter values that make your observed sample most plausible under a chosen probability model. The calculator applies closed-form MLE formulas for the supported distributions.

2) Why do I need to choose a distribution first?

MLE depends on the model behind the data. Normal, Exponential, Poisson, and Bernoulli data each use different likelihood functions, so the parameter formulas also differ.

3) What kind of data works for the Normal model?

Use continuous data that can vary around a central mean. The page estimates both the mean and standard deviation from the sample and plots the fitted density.

4) When should I use the Exponential model?

Use it for strictly positive waiting times or lifetimes when events occur continuously and memoryless behavior is a reasonable approximation.

5) What kind of data fits the Poisson model?

Poisson is appropriate for non-negative integer event counts measured over equal intervals, such as arrivals per minute, defects per sheet, or calls per hour.

6) Why does Bernoulli require only zeros and ones?

A Bernoulli variable has only two outcomes: failure and success. The MLE then becomes the observed success proportion, which is the sample mean of those binary values.

7) What do AIC and BIC tell me?

They measure fit while penalizing extra parameters. Lower values usually indicate a better model when you compare several candidate distributions on the same sample.

8) Are the confidence intervals exact?

They are approximate intervals based on standard large-sample formulas. For small samples or boundary cases, exact or bootstrap intervals may be more reliable.