1. What is the Cumulative Distribution Function?

The Cumulative Distribution Function (CDF) gives the probability that a random variable takes a value less than or equal to a specific point. It's the integral of the probability density function (PDF) from negative infinity to x.

2. How Does the Calculator Work?

The calculator uses the CDF formula:

Where:

• \( f(t) \) — Probability density function
• \( x \) — Value at which to evaluate the CDF

Explanation: The CDF represents the area under the PDF curve from negative infinity to the point x.

3. Importance of CDF Calculation

Details: The CDF is fundamental in probability theory and statistics, used for calculating probabilities, determining percentiles, and statistical hypothesis testing.

4. Using the Calculator

Tips: Enter the probability density function formula (e.g., "exp(-t^2/2)/sqrt(2*pi)" for standard normal) and the x value where you want to evaluate the CDF.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between PDF and CDF?
A: The PDF gives the probability density at a point, while the CDF gives the accumulated probability up to that point.

Q2: What are the properties of a CDF?
A: CDFs are always non-decreasing, right-continuous functions with limits of 0 at -∞ and 1 at +∞.

Q3: Can I use this for discrete distributions?
A: For discrete distributions, the CDF is a sum rather than an integral of the probability mass function.

Q4: How accurate is this calculator?
A: Accuracy depends on the numerical integration method used for the specific PDF.

Q5: What common distributions have known CDFs?
A: Normal, exponential, uniform, and many other distributions have well-known CDF formulas.