Probability Distribution Calculator

About Probability Distribution Calculator

Welcome to our Probability Distribution Calculator, a comprehensive tool designed to compute probabilities, cumulative probabilities, and quantiles for various probability distributions with detailed step-by-step solutions. This calculator is ideal for students, teachers, and anyone working with probability and statistics.

Features of the Probability Distribution Calculator

• Step-by-Step Solutions: Understand each step involved in probability computations.
• User-Friendly Interface: Input parameters easily and get instant results.
• Supports Multiple Distributions: Normal, Binomial, Poisson, Exponential, and Uniform distributions.

Understanding Probability Distributions

Probability distributions describe how probabilities are distributed over the values of a random variable. Below are the formulas and comparisons for each supported distribution.

Normal Distribution

The Normal distribution is a continuous probability distribution characterized by its mean \( \mu \) and standard deviation \( \sigma \).

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

Binomial Distribution

The Binomial distribution is a discrete probability distribution representing the number of successes in \( n \) independent Bernoulli trials with probability \( p \) of success.

• PMF: \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \)
• CDF: \( F(k) = P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1 - p)^{n - i} \)
• Quantile Function: Inverse of the CDF for given \( p \).

Poisson Distribution

The Poisson distribution is a discrete probability distribution expressing the probability of a given number of events occurring in a fixed interval of time or space.

• PMF: \( P(X = k) = \dfrac{e^{-\lambda} \lambda^{k}}{k!} \)
• CDF: \( F(k) = P(X \leq k) = e^{-\lambda} \sum_{i=0}^{k} \dfrac{\lambda^{i}}{i!} \)
• Quantile Function: Inverse of the CDF for given \( p \).

Exponential Distribution

The Exponential distribution is a continuous probability distribution commonly used to model the time between independent events that happen at a constant average rate.

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

Uniform Distribution

The Uniform distribution is a continuous probability distribution where all intervals of the same length are equally probable within the interval \( [a, b] \).

• PDF: \( f(x) = \dfrac{1}{b - a} \) for \( a \leq x \leq b \)
• CDF: \( F(x) = \dfrac{x - a}{b - a} \) for \( a \leq x \leq b \)
• Quantile Function: \( x = a + p(b - a) \)

Comparisons and Applications

Each distribution serves different purposes and models different types of data:

• Normal Distribution: Used for continuous data that clusters around a mean. Applicable in natural and social sciences.
• Binomial Distribution: Models the number of successes in a fixed number of independent Bernoulli trials. Used in quality control and genetics.
• Poisson Distribution: Suitable for counting the number of events in a fixed interval. Used in telecommunications and traffic engineering.
• Exponential Distribution: Models the time between events in a Poisson process. Used in reliability engineering and queuing theory.
• Uniform Distribution: Represents equal probability over an interval. Used in simulations and random sampling.

How to Use the Probability Distribution Calculator

• Select the distribution you wish to use.
• Select the calculation type: PDF/PMF, CDF, or Quantile (Inverse CDF).
• Enter the required parameters and the value or probability.
• Click on "Compute" to process your inputs.
• View the result along with detailed step-by-step solutions.

Additional Resources

• Probability Distribution - Wikipedia
• Statistics and Probability - Khan Academy
• MIT OpenCourseWare - Introduction to Probability and Statistics

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