Probability Density Function Calculator

Gaussian PDF Formula:

\[ PDF = \frac{1}{\sigma \sqrt{2\pi}} \times e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]

PDF Value:

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1. What is the Probability Density Function?

The Probability Density Function (PDF) describes the relative likelihood for a continuous random variable to take on a given value. The Gaussian (normal) distribution is one of the most important probability distributions in statistics.

2. How Does the Calculator Work?

The calculator uses the Gaussian PDF formula:

\[ PDF = \frac{1}{\sigma \sqrt{2\pi}} \times e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]

Where:

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

Explanation: The formula calculates the probability density at point x for a normal distribution with given mean and standard deviation.

3. Importance of PDF Calculation

Details: PDF calculations are fundamental in statistics, probability theory, and various scientific fields. They help understand distribution characteristics, calculate probabilities for intervals, and perform statistical inference.

4. Using the Calculator

Tips: Enter the mean (μ), standard deviation (σ > 0), and the x value at which you want to evaluate the PDF. All values must be valid real numbers.

5. Frequently Asked Questions (FAQ)

Q1: What does the PDF value represent?
A: The PDF value represents the relative likelihood of the random variable taking on a specific value. For continuous distributions, the probability at any single point is zero; probabilities are calculated over intervals.

Q2: Why is the Gaussian distribution so important?
A: The Gaussian distribution appears naturally in many phenomena (Central Limit Theorem) and is widely used in statistical modeling, hypothesis testing, and machine learning.

Q3: Can PDF values be greater than 1?
A: Yes, PDF values can be greater than 1. The important property is that the total area under the PDF curve equals 1, not that individual values are less than 1.

Q4: What's the difference between PDF and PMF?
A: PDF is for continuous random variables, while PMF (Probability Mass Function) is for discrete random variables. PDF gives density, PMF gives actual probabilities.

Q5: How is the standard deviation related to the PDF shape?
A: Larger standard deviation means wider, flatter distribution. Smaller standard deviation means narrower, taller distribution centered around the mean.