Probability Distribution Calculator

Welcome to the Probability Distribution Calculator, a comprehensive statistical tool for computing probabilities, cumulative probabilities (CDF), and quantiles (inverse CDF) for various probability distributions. Whether you're a student learning statistics, a researcher analyzing data, or a professional working with statistical models, this calculator provides detailed step-by-step solutions and interactive visualizations to help you understand probability distributions.

Supported Probability Distributions

This calculator supports seven commonly used probability distributions, each suited for different types of random phenomena:

Understanding PDF, CDF, and Quantile Functions

Probability Density/Mass Function (PDF/PMF)

The PDF (for continuous distributions) or PMF (for discrete distributions) gives the relative likelihood of a random variable taking a specific value. For continuous distributions, the PDF value itself is not a probability but a density—probabilities are found by integrating the PDF over an interval.

Cumulative Distribution Function (CDF)

The CDF, denoted F(x), gives the probability that a random variable X is less than or equal to a value x. This is written as P(X ≤ x). The CDF always increases from 0 to 1 as x increases.

Quantile Function (Inverse CDF)

The quantile function (also called percent-point function or inverse CDF) finds the value x for which P(X ≤ x) = p. It answers: "What value is exceeded by only (1-p)×100% of the distribution?" This is essential for finding critical values in hypothesis testing.

Distribution Formulas

Normal Distribution

The Normal (Gaussian) distribution is symmetric and bell-shaped, characterized by mean μ (center) and standard deviation σ (spread).

• PDF: \( f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)
• CDF: \( F(x) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right] \)
• Quantile: \( x = \mu + \sigma \cdot \Phi^{-1}(p) \)

Binomial Distribution

Models the number of successes in n independent trials, each with success probability p.

• PMF: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \)
• CDF: \( F(k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{n-i} \)

Poisson Distribution

Models the number of events in a fixed interval when events occur at a constant average rate λ.

• PMF: \( P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \)
• CDF: \( F(k) = e^{-\lambda} \sum_{i=0}^{k} \frac{\lambda^i}{i!} \)

Exponential Distribution

Models the time between events in a Poisson process with rate λ.

• PDF: \( f(x) = \lambda e^{-\lambda x} \) for x ≥ 0
• CDF: \( F(x) = 1 - e^{-\lambda x} \)
• Quantile: \( x = -\frac{\ln(1-p)}{\lambda} \)

Chi-Square Distribution

Arises in statistics as the sum of squared standard normal variables. Used in hypothesis testing and confidence intervals for variance.

• PDF: \( f(x) = \frac{x^{k/2-1} e^{-x/2}}{2^{k/2} \Gamma(k/2)} \) for x > 0

Student's t Distribution

Similar to Normal but with heavier tails. Used for inference about population means when sample size is small or population variance is unknown.

• PDF: \( f(x) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\left(\frac{\nu}{2}\right)} \left(1+\frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}} \)

How to Use This Calculator

1. Select a distribution: Click on the distribution card that matches your data or problem. Each card shows the distribution type (continuous or discrete).
2. Choose calculation type: Select PDF/PMF for probability at a point, CDF for cumulative probability, or Quantile to find a value for a given probability.
3. Enter parameters: Input the distribution parameters. The form dynamically shows only relevant parameters for your chosen distribution.
4. Enter the value or probability: For PDF/CDF, enter the x-value (or k for discrete). For Quantile, enter a probability between 0 and 1.
5. Review results: Examine the computed result, step-by-step mathematical derivation, and interactive distribution visualization.

Frequently Asked Questions

What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different possible outcomes for a random variable. It can be discrete (like Binomial or Poisson) for countable outcomes, or continuous (like Normal or Exponential) for outcomes that can take any value within a range.

What is the difference between PDF and CDF?

PDF (Probability Density Function) or PMF (Probability Mass Function) gives the probability density at a specific point. For discrete distributions, PMF gives the exact probability P(X=k). CDF (Cumulative Distribution Function) gives the probability that the random variable is less than or equal to a value: P(X≤x). CDF is the cumulative sum/integral of PDF/PMF.

When should I use Normal distribution?

Normal distribution is appropriate for continuous data that is symmetrically distributed around a mean value. It's commonly used for phenomena like heights, test scores, measurement errors, and many biological variables. The Central Limit Theorem states that sample means tend toward normal distribution regardless of population distribution.

What is a quantile function?

The quantile function (also called inverse CDF or percent-point function) finds the value x such that P(X≤x) = p for a given probability p. For example, the 95th percentile (p=0.95) of a distribution is the value below which 95% of observations fall.

How do I choose between different distributions?

Choose based on your data characteristics: Normal for symmetric continuous data around a mean; Binomial for counting successes in fixed trials; Poisson for counting rare events in a fixed interval; Exponential for time between events; Uniform for equal probability across a range; Chi-Square for variance testing; Student's t for small samples with unknown population variance.

Additional Resources

• Probability Distribution - Wikipedia
• Normal Distribution - Wikipedia
• Statistics and Probability - Khan Academy