Normal Probability Density Function Calculator

Normal PDF Formula:

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

Probability Density:

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1. What is the Normal Probability Density Function?

The Normal Probability Density Function (PDF) describes the relative likelihood of a random variable taking on a given value in a normal distribution. It's characterized by its bell-shaped curve and is defined by two parameters: mean (μ) and standard deviation (σ).

2. How Does the Calculator Work?

The calculator uses the normal PDF formula:

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

Where:

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

Explanation: The equation calculates the height of the normal curve at point x, showing how likely values are near x.

3. Importance of Normal PDF

Details: The normal distribution is fundamental in statistics, describing many natural phenomena. The PDF is used in probability calculations, statistical inference, and quality control.

4. Using the Calculator

Tips: Enter the mean (μ), standard deviation (σ > 0), and the x value where you want to evaluate the PDF. The calculator will return the probability density at that point.

5. Frequently Asked Questions (FAQ)

Q1: What does the probability density value mean?
A: It represents the relative likelihood of the random variable being near x. For continuous distributions, probability at a single point is technically zero.

Q2: What's the difference between PDF and CDF?
A: PDF gives the density at a point, while CDF (Cumulative Distribution Function) gives the probability of being less than or equal to a value.

Q3: What's the standard normal distribution?
A: A special case where μ = 0 and σ = 1. The formula simplifies to \( \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \).

Q4: Why is the normal distribution so important?
A: Due to the Central Limit Theorem, sums of many random variables tend toward normal distribution, making it fundamental in statistics.

Q5: Can the PDF value be greater than 1?
A: Yes, PDF values can be >1. The area under the entire curve must be 1, but the height at any point can exceed 1, especially with small σ.