https://github.com/amirjahantab/probability_distribution_python

common Probability Distributions
https://github.com/amirjahantab/probability_distribution_python

mathematics normal-distribution probability probability-distribution

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common Probability Distributions

• Host: GitHub
• URL: https://github.com/amirjahantab/probability_distribution_python
• Owner: amirjahantab
• Created: 2024-06-06T14:23:36.000Z (8 months ago)
• Default Branch: main
• Last Pushed: 2024-06-23T09:27:55.000Z (8 months ago)
• Last Synced: 2024-11-12T18:46:00.432Z (3 months ago)
• Topics: mathematics, normal-distribution, probability, probability-distribution
• Language: Jupyter Notebook
• Homepage:
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• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
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  • Readme: README.md

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README

        
# Probability Distributions

## Discrete Uniform Distribution

The discrete uniform distribution assigns equal probability to all integers between two specified values $a$ and $b$.

**Formula:**

$\ P(X=x) = \frac{1}{b-a+1} \$

## Bernoulli Distribution

The Bernoulli distribution is a discrete distribution with only two possible outcomes, usually coded as 0 and 1. It is parameterized by $p$, the probability of success (1).

**Formula:**

$\ P(X=x) = \$




    $\ p \$

     $\ if \$

     $\ n = 1 \$




      $\ 1-p \$

     $\ if \$

     $\ n = 0 \$

## Binomial Distribution

The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials.

**Formula:**

$\ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \$

## Poisson Distribution

The Poisson distribution models the number of events occurring within a fixed interval of time or space.

**Formula:**

$\ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \$

## Geometric Distribution

The geometric distribution models the number of trials until the first success in a sequence of independent Bernoulli trials.

**Formula:**

$\ P(X=k) = (1-p)^{k-1} p \$

## Continuous Uniform Distribution

The continuous uniform distribution assigns equal probability to all points in the interval $[a, b]$.

**Formula:**

$\ f(x) = \frac{1}{b-a} \$

## Normal Distribution

The normal distribution is a continuous distribution characterized by its mean $\mu$ and standard deviation $\sigma$.

**Formula:**

$\ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \$

## Exponential Distribution

The exponential distribution models the time between events in a Poisson process.

**Formula:**

$\ f(x; \lambda) = \lambda e^{-\lambda x} \$

## Chi-Square (χ²) Distribution

The chi-square distribution is a special case of the gamma distribution, often used in hypothesis testing.

**Formula:**

$\ f(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{(k/2)-1} e^{-x/2} \$

## Gamma Distribution

The gamma distribution generalizes the exponential distribution with shape parameter $k$ and scale parameter $\theta$.

**Formula:**

$\ f(x; k, \theta) = \frac{x^{k-1} e^{-x/\theta}}{\theta^k \Gamma(k)} \$