1. What is Expected Value?

The expected value (E[X]) of a continuous random variable is a measure of the center of its distribution, representing the long-run average value of repetitions of the experiment it represents.

2. How Does the Calculator Work?

The calculator uses the expected value formula:

Where:

• \( E[X] \) — Expected value
• \( f(x) \) — Probability density function
• \( x \) — Value of the random variable

Explanation: The integral calculates the weighted average of all possible values of the random variable, where the weights are given by the probability density function.

3. Importance of Expected Value

Details: Expected value is fundamental in probability theory and statistics, used in decision making, risk assessment, and various applications across finance, insurance, and engineering.

4. Using the Calculator

Tips: Enter a valid probability density function f(x) and the bounds of integration. The function must be non-negative and integrate to 1 over its domain.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can I enter?
A: You can enter any continuous probability density function, such as normal, uniform, or exponential distributions.

Q2: How accurate is the calculation?
A: Accuracy depends on the numerical integration method used. More complex functions may require more computational resources.

Q3: What if my function is not normalized?
A: The calculator assumes the input is a proper probability density function (integrates to 1). Non-normalized inputs will give incorrect results.

Q4: Can I use this for discrete distributions?
A: No, this calculator is specifically for continuous distributions. For discrete distributions, use the summation formula.

Q5: What about infinite bounds?
A: For practical computation, you must specify finite bounds. Choose bounds that capture most of the probability mass.