Gaussian Distribution Calculator

Gaussian Distribution Formula:

\[ PDF = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) \]

Probability Density Function (PDF):

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1. What is the Gaussian Distribution?

The Gaussian distribution, also known as the normal distribution, is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

2. How Does the Calculator Work?

The calculator uses the Gaussian PDF formula:

\[ PDF = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) \]

Where:

\( x \) — The value at which to evaluate the distribution
\( \mu \) — The mean of the distribution
\( \sigma \) — The standard deviation of the distribution
\( \pi \) — Mathematical constant pi (approximately 3.14159)
\( e \) — Base of the natural logarithm (approximately 2.71828)

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

3. Importance of Gaussian Distribution

Details: The Gaussian distribution is fundamental in statistics and appears in many natural phenomena. It's used in hypothesis testing, confidence intervals, and as an approximation for various probability distributions.

4. Using the Calculator

Tips: Enter the value (x), mean (μ), and standard deviation (σ). Standard deviation must be greater than zero. The calculator will compute the probability density at the specified point.

5. Frequently Asked Questions (FAQ)

Q1: What does the PDF value represent?
A: The PDF value represents the relative likelihood that a random variable would equal the specified value x. It's not a probability but a density.

Q2: What is the difference between PDF and CDF?
A: PDF gives the density at a specific point, while CDF (Cumulative Distribution Function) gives the probability that a random variable is less than or equal to a specific value.

Q3: What are typical values for mean and standard deviation?
A: The mean determines the center of the distribution, while the standard deviation controls the spread. Common values depend on the specific application.

Q4: Can standard deviation be zero?
A: No, standard deviation must be greater than zero for the Gaussian distribution to be defined. A standard deviation of zero would represent a degenerate distribution.

Q5: Where is the Gaussian distribution commonly used?
A: It's used in natural sciences, social sciences, finance, engineering, and many other fields to model real-valued random variables whose distributions are not known.