1. What is Probability Density Function?

The Probability Density Function (PDF) describes the relative likelihood for a continuous random variable to take on a given value. It represents the derivative of the cumulative distribution function and is fundamental in probability theory and statistics.

2. How Does the Calculator Work?

The calculator uses the PDF formula:

Where:

• \( P \) — Probability value between 0 and 1
• \( a \) — Lower bound value
• \( b \) — Upper bound value
• \( X \) — Random variable

Explanation: The PDF represents the limit of the probability that X falls in the interval [a, b] divided by the length of the interval, as the interval length approaches zero.

3. Importance of PDF Calculation

Details: PDF is essential for understanding probability distributions, calculating probabilities for continuous variables, and is fundamental in statistical analysis, machine learning, and various scientific fields.

4. Using the Calculator

Tips: Enter the probability value (between 0 and 1), and the lower and upper bounds (a and b). The values of a and b must be different for the calculation to be valid.

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 does the area under a PDF curve represent?
A: The area under the PDF curve between two points represents the probability that the random variable falls within that interval.

Q4: Are all PDFs symmetric?
A: No, PDFs can have various shapes including normal (symmetric), exponential (right-skewed), and many other distributions.

Q5: How is PDF related to CDF?
A: The Cumulative Distribution Function (CDF) is the integral of the PDF, representing the probability that a random variable is less than or equal to a certain value.