Probability Density Function Calculation

Probability Density Function:

\[ f(x) = \frac{dF(x)}{dx} \]

Probability Density Function f(x):

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1. What is a Probability Density Function?

A Probability Density Function (PDF) describes the relative likelihood for a continuous random variable to take on a given value. It is the derivative of the cumulative distribution function (CDF).

2. How Does the Calculator Work?

The calculator uses differentiation:

\[ f(x) = \frac{dF(x)}{dx} \]

Where:

\( f(x) \) — Probability density function
\( F(x) \) — Cumulative distribution function
\( \frac{d}{dx} \) — Derivative with respect to x

Explanation: The PDF represents the rate of change of the cumulative distribution function.

3. Importance of PDF Calculation

Details: PDFs are fundamental in probability theory and statistics, used to determine probabilities for continuous random variables and to calculate expected values and variances.

4. Using the Calculator

Tips: Enter the cumulative distribution function F(x) and the variable with respect to which you want to differentiate. The calculator will compute the corresponding probability density function.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between PDF and PMF?
A: PDF is for continuous random variables, while Probability Mass Function (PMF) is for discrete random variables.

Q2: Can PDF values be greater than 1?
A: Yes, PDF values can be greater than 1, but the integral over the entire space must equal 1.

Q3: What are some common PDFs?
A: Normal distribution, exponential distribution, uniform distribution, and beta distribution are common examples.

Q4: How is PDF related to CDF?
A: PDF is the derivative of CDF, and CDF is the integral of PDF.

Q5: What are the properties of a valid PDF?
A: A valid PDF must be non-negative everywhere, and its integral over the entire space must equal 1.