1. What is the Gaussian Probability Density Function?

The Gaussian (or normal) probability density function describes the relative likelihood of a continuous random variable taking on a particular value. It's characterized by its bell-shaped curve and is fundamental in statistics.

2. How Does the Calculator Work?

The calculator uses the Gaussian PDF formula:

Where:

• \( \mu \) — Mean of the distribution
• \( \sigma \) — Standard deviation (>0)
• \( x \) — Value at which to evaluate the function
• \( \pi \) — Mathematical constant pi (~3.14159)
• \( e \) — Euler's number (~2.71828)

Explanation: The function gives the relative likelihood of observing a particular value in a normally distributed population.

3. Importance of Gaussian Distribution

Details: The normal distribution is fundamental in statistics, appearing naturally in many situations (central limit theorem). It's used in hypothesis testing, quality control, and risk assessment.

4. Using the Calculator

Tips: Enter the mean (μ), standard deviation (σ > 0), and the x value where you want to evaluate the density. All values are dimensionless.

5. Frequently Asked Questions (FAQ)

Q1: What does the probability density value mean?
A: It's the relative likelihood per unit of the random variable. For exact probabilities, you need to integrate over an interval.

Q2: What's the difference between PDF and PMF?
A: PDF is for continuous variables, PMF is for discrete variables. PDF gives density, not direct probability.

Q3: What happens when standard deviation approaches zero?
A: The curve becomes infinitely tall and narrow, approaching a Dirac delta function.

Q4: Can the PDF value be greater than 1?
A: Yes, PDF values can be >1. The important property is that the total area under the curve is 1.

Q5: How is this related to the standard normal distribution?
A: When μ=0 and σ=1, it's the standard normal distribution. Any normal distribution can be standardized.