Probability Distribution Calculator

Compute probabilities, cumulative probabilities, and quantiles for various probability distributions with detailed step-by-step solutions!

Compute Probability Distribution:

Try the following examples:

[Normal Distribution CDF Example] [Binomial Distribution PMF Example] [Poisson Distribution CDF Example] [Exponential Distribution Quantile Example] [Uniform Distribution PDF Example]

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Value/Probability:

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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

Reference this content, page, or tool as:

"Probability Distribution Calculator" at https://MiniWebtool.com/probability-distribution-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Nov 22, 2024

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About Probability Distribution Calculator

Features of the Probability Distribution Calculator

Understanding Probability Distributions

Normal Distribution

Binomial Distribution

Poisson Distribution

Exponential Distribution

Uniform Distribution

Comparisons and Applications

How to Use the Probability Distribution Calculator

Additional Resources

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