1. What is Probability Density Function Mean?

The mean of a probability density function represents the expected value or average outcome of a continuous random variable. It provides the central tendency measure for the probability distribution.

2. How Does the Calculator Work?

The calculator uses the mean formula:

Where:

• \( \mu \) — Mean or expected value
• \( x \) — Random variable value
• \( f(x) \) — Probability density function at x

Explanation: The calculator approximates the integral using discrete x and f(x) values provided by the user, calculating the sum of x*f(x) products.

3. Importance of Mean Calculation

Details: Calculating the mean of a probability density function is essential for understanding the central tendency of continuous distributions, statistical analysis, and probability theory applications.

4. Using the Calculator

Tips: Enter x values and corresponding f(x) PDF values as comma-separated lists. Ensure both lists have the same number of values and represent valid probability densities.

5. Frequently Asked Questions (FAQ)

Q1: What is a probability density function?
A: A PDF describes the relative likelihood for a continuous random variable to take on a given value. The area under the curve equals 1.

Q2: How is this different from discrete mean calculation?
A: For discrete distributions, we use summation. For continuous distributions, we use integration to find the mean.

Q3: What are common PDFs where mean is calculated?
A: Normal distribution, exponential distribution, uniform distribution, and many other continuous probability distributions.

Q4: Why is numerical integration used here?
A: Since users provide discrete values, we approximate the integral using the sum of x*f(x) products, which works well for evenly spaced samples.

Q5: What if my PDF doesn't integrate to 1?
A: The calculator assumes valid probability densities. If your f(x) values don't sum to approximately 1 (considering spacing), the mean calculation may be inaccurate.