1. What is Likelihood?

Likelihood is a fundamental concept in statistics that measures how well a set of parameters explains observed data. Unlike probability, which predicts outcomes given parameters, likelihood assesses parameter plausibility given observed data.

2. How Does the Calculator Work?

The calculator uses the likelihood function:

Where:

• \( L(\theta) \) — Likelihood function
• \( \theta \) — Parameter vector
• \( n \) — Number of data points
• \( P(data_i \mid \theta) \) — Probability of data point i given parameters θ
• \( \prod \) — Product operator (multiplication across all data points)

Explanation: The likelihood is the product of probabilities of observing each data point given the parameters. Higher likelihood values indicate better parameter fit.

3. Importance of Likelihood Calculation

Details: Likelihood forms the basis for maximum likelihood estimation (MLE), hypothesis testing, model selection, and Bayesian statistics. It's essential for parameter estimation and statistical inference.

4. Using the Calculator

Tips: Enter the number of data points, individual probability (0-1), and select the appropriate probability distribution. The calculator computes the joint likelihood across all data points.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between likelihood and probability?
A: Probability predicts data given parameters, while likelihood assesses parameters given data. Probability sums to 1, likelihood does not.

Q2: Why use product instead of sum in likelihood?
A: The product assumes data independence. For independent observations, joint probability is the product of individual probabilities.

Q3: What is maximum likelihood estimation (MLE)?
A: MLE finds parameter values that maximize the likelihood function, providing the most plausible parameters given observed data.

Q4: When should I use log-likelihood?
A: Use log-likelihood for numerical stability with many data points, as it converts products to sums and handles very small numbers better.

Q5: Can likelihood be greater than 1?
A: Yes, since likelihood isn't a probability measure. It's a relative measure used for comparison, not absolute interpretation.