1. What is Inverse Distribution?

The inverse distribution function (also called quantile function) takes a probability and returns the value at which the cumulative distribution function equals that probability. It's essential for finding critical values in hypothesis testing and constructing confidence intervals.

2. How the Calculator Works

The calculator uses the general formula:

Where:

• \( F^{-1} \) — Inverse cumulative distribution function
• \( p \) — Probability value (0 to 1)
• \( x \) — The value at which P(X ≤ x) = p

Supported Distributions: Normal, t, Chi-squared, and F distributions.

3. Applications of Inverse Distribution

Details: Inverse distribution is used in statistical hypothesis testing to find critical values, in risk management to calculate Value at Risk (VaR), and in simulation studies for random number generation.

4. Using the Calculator

Steps: Select distribution type, enter probability (0-1), and provide required degrees of freedom. The calculator will return the corresponding x value.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between PDF and inverse CDF?
A: PDF gives probability density at a point, while inverse CDF gives the point at which the cumulative probability equals a specific value.

Q2: Why do I need degrees of freedom for some distributions?
A: Distributions like t, chi-squared, and F have shape parameters (degrees of freedom) that affect their form.

Q3: Can I use this for non-standard normal distributions?
A: For N(μ,σ²), first find z for N(0,1), then transform: x = μ + zσ.

Q4: What if I need the two-tailed inverse?
A: For two-tailed tests, use p/2 for one tail and 1-p/2 for the other.

Q5: How precise are these calculations?
A: Results are accurate to 4 decimal places, suitable for most statistical applications.