feat: rework desctiption

Author: peplxx

distributions/base.py

@@ -18,6 +18,6 @@ def get_formula(self):
         pass
 
     @abstractmethod
-    def get_properties(self):
+    def get_properties(self, st):
         """Return distribution properties"""
-        pass
+        pass

distributions/continuous/__init__.py

@@ -2,5 +2,6 @@
 from .mutivariative_normal import MultivariateNormalDistribution
 from .chi_squared import ChiSquaredDistribution
 from .uniform import UniformDistribution
+from .exponential import ExponentialDistribution
 
-__all__ = ['NormalDistribution', 'MultivariateNormalDistribution', 'ChiSquaredDistribution', 'UniformDistribution']
+__all__ = ['NormalDistribution', 'MultivariateNormalDistribution', 'ChiSquaredDistribution', 'UniformDistribution', 'ExponentialDistribution']

distributions/continuous/chi_squared.py

@@ -18,11 +18,32 @@ def plot(self, ax):
     def get_formula(self):
         return r'f(x) = \frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}e^{-x/2}'
 
-    def get_properties(self):
-        return """
-        - Support is x > 0 
-        - Mean equals degrees of freedom (k)
-        - Variance equals 2k
-        - Right-skewed distribution
-        - Sum of k squared standard normal variables
-        """
+    def get_properties(self, st):
+        st.write("""
+        The **Chi-squared distribution** is a continuous probability distribution that is commonly used in statistical inference.
+        It is characterized by one parameter:
+        - **k (degrees of freedom)**: determines the shape of the distribution
+        
+        Key Properties:
+        - **Support is x > 0** (non-negative values only)
+        - **Mean equals degrees of freedom (k)**
+        - **Variance equals 2k**
+        - **Right-skewed distribution**
+        - **Sum of k squared standard normal variables**
+        
+        The probability density function (PDF) is given by:
+        """)
+        
+        st.latex(r'f(x) = \frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}e^{-x/2}')
+        
+        st.write("""
+        Common applications include:
+        - **Goodness-of-fit tests**
+        - **Testing independence in contingency tables**
+        - **Confidence intervals for population variance**
+        - **Component in other distributions** (e.g., F-distribution)
+        
+        Important links:
+        - [**Properties**](https://en.wikipedia.org/wiki/Chi-square_distribution#Properties)
+        - [**Applications**](https://en.wikipedia.org/wiki/Chi-square_distribution#Applications)
+        """)

distributions/continuous/exponential.py

@@ -0,0 +1,56 @@
+import numpy as np
+from scipy.stats import expon
+import streamlit as st
+from ..base import Distribution
+
+class ExponentialDistribution(Distribution):
+    def get_parameters(self):
+        self.rate = st.slider('Rate parameter (λ)', 0.1, 5.0, 1.0, 0.1)
+        return {'rate': self.rate}
+
+    def plot(self, ax):
+        x = np.linspace(0, 5/self.rate, 200)
+        y = self.rate * np.exp(-self.rate * x)
+        ax.plot(x, y)
+        ax.set_xlabel('x')
+        ax.set_ylabel('Probability Density')
+
+    def get_formula(self):
+        return r'f(x) = \lambda e^{-\lambda x}'
+
+    def get_properties(self, st):
+        st.write("""
+        The **Exponential Distribution** is a continuous probability distribution that describes the time between events in a Poisson point process.
+        It is characterized by one parameter:
+        - **λ (rate parameter)**: determines the rate of decay
+        
+        Key Properties:
+        - **Support is x ≥ 0** (non-negative values only)
+        - **Mean = 1/λ**
+        - **Variance = 1/λ²**
+        - **Memoryless property**: P(X > s + t | X > s) = P(X > t)
+        - **Maximum entropy** distribution for a given mean
+        
+        The probability density function (PDF) is given by:
+        """)
+        
+        st.latex(r'f(x) = \lambda e^{-\lambda x} \text{ for } x \geq 0')
+        
+        st.write("""
+        The cumulative distribution function (CDF) is:
+        """)
+        
+        st.latex(r'F(x) = 1 - e^{-\lambda x} \text{ for } x \geq 0')
+        
+        st.write("""
+        Common applications:
+        - **Lifetime/survival analysis**
+        - **Time between events**
+        - **Reliability engineering**
+        - **Queueing theory**
+        
+        Important links:
+        - [**Properties**](https://en.wikipedia.org/wiki/Exponential_distribution#Properties)
+        - [**Applications**](https://en.wikipedia.org/wiki/Exponential_distribution#Applications)
+        - [**Relationship to other distributions**](https://en.wikipedia.org/wiki/Exponential_distribution#Related_distributions)
+        """)

distributions/continuous/mutivariative_normal.py

@@ -37,11 +37,40 @@ def plot(self, ax):
     def get_formula(self):
         return r'f(x) = \frac{1}{2\pi|\Sigma|^{1/2}} \exp\left(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)'
 
-    def get_properties(self):
-        return """
-        - Generalizes univariate normal to multiple dimensions
-        - Characterized by mean vector and covariance matrix
-        - Elliptical contours of constant density
-        - Marginal and conditional distributions are normal
-        - Linear combinations are normally distributed
-        """
+    def get_properties(self, st):
+        st.write("""
+        The **Multivariate Normal Distribution** is a generalization of the univariate normal distribution to multiple dimensions.
+        It is characterized by two parameters:
+        - **μ (mean vector)**: determines the center of the distribution
+        - **Σ (covariance matrix)**: determines the shape, spread and orientation of the distribution
+        
+        Key Properties:
+        - **Generalizes univariate normal to multiple dimensions**
+        - **Characterized by mean vector and covariance matrix**
+        - **Elliptical contours of constant density**
+        - **Marginal and conditional distributions are normal**
+        - **Linear combinations are normally distributed**
+        
+        The probability density function (PDF) is given by:
+        """)
+        
+        st.latex(r'f(x) = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}} \exp\left(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)')
+        
+        st.write("""
+        where:
+        - **x** is an n-dimensional vector
+        - **μ** is the mean vector
+        - **Σ** is the covariance matrix
+        - **|Σ|** is the determinant of Σ
+        
+        The cumulative distribution function (CDF) does not have a closed form for n>1, but can be expressed as:
+        """)
+        
+        st.latex(r'F(x) = \int_{-\infty}^{x_1}\cdots\int_{-\infty}^{x_n} f(t_1,\ldots,t_n)dt_1\cdots dt_n')
+        
+        st.write("""
+        Important links:
+        - [**Maximum Likelihood Estimation**](https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Maximum_likelihood_estimation)
+        - [**Properties**](https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Properties)
+        - [**Applications**](https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Applications)
+        """)

distributions/continuous/normal.py

@@ -19,10 +19,35 @@ def plot(self, ax):
     def get_formula(self):
         return r'f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}'
 
-    def get_properties(self):
-        return """
-        - Symmetric about mean
-        - 68-95-99.7 rule applies
-        - Bell-shaped curve
-        - Infinite support
-        """
+    def get_properties(self, st):
+        st.write("""
+        The **Normal** (or **Gaussian**) distribution is one of the most important probability distributions in statistics. 
+        It is characterized by two parameters:
+        - **μ (mean)**: determines the center of the distribution
+        - **σ (standard deviation)**: determines the spread/width of the distribution
+        
+        Key features:
+        - **Symmetric bell-shaped curve centered at the mean**
+        - **About 68% of data falls within 1 standard deviation of mean**
+        - **About 95% of data falls within 2 standard deviations**
+        - **About 99.7% of data falls within 3 standard deviations**
+        
+        The probability density function (PDF) gives the relative likelihood of a random variable taking on a specific value:
+        """)
+        
+        st.latex(r'f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}')
+        
+        st.write("""
+        The cumulative distribution function (CDF) gives the probability that a random variable takes a value less than or equal to x:
+        """)
+        
+        st.latex(r'F(x) = \frac{1}{2}[1 + \text{erf}(\frac{x-\mu}{\sigma\sqrt{2}})]')
+        
+        st.write("""
+        The normal distribution is extremely important because:
+        - **It appears naturally in many phenomena** ([Central Limit Theorem](https://en.wikipedia.org/wiki/Central_limit_theorem))
+        - **It is easy to work with mathematically**
+        - **Many statistical methods assume** [normality](https://en.wikipedia.org/wiki/Normal_distribution#Normality_tests)
+        - **It is the** [maximum entropy](https://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution) **distribution for given mean and variance**
+        """)
+        return

distributions/continuous/uniform.py

@@ -21,11 +21,41 @@ def plot(self, ax):
     def get_formula(self):
         return r'f(x) = \frac{1}{b-a} \text{ for } a \leq x \leq b'
 
-    def get_properties(self):
-        return """
-        - Constant probability density over interval [a,b]
-        - Mean = (a+b)/2
-        - Variance = (b-a)²/12
-        - Maximum entropy continuous distribution for interval [a,b]
-        - All points in interval equally likely
-        """
+    def get_properties(self, st):
+        st.write("""
+        The **Uniform Distribution** is the simplest continuous probability distribution.
+        It is characterized by two parameters:
+        - **a (lower bound)**: minimum value of the interval
+        - **b (upper bound)**: maximum value of the interval
+        
+        Key Properties:
+        - **Constant probability density** of 1/(b-a) over interval [a,b]
+        - **Zero probability density** outside interval [a,b]
+        - **Mean = (a+b)/2**
+        - **Variance = (b-a)²/12**
+        - **Maximum entropy** continuous distribution for a bounded interval
+        - **All points in interval are equally likely**
+        
+        The probability density function (PDF) is given by:
+        """)
+        
+        st.latex(r'f(x) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}')
+        
+        st.write("""
+        The cumulative distribution function (CDF) is:
+        """)
+        
+        st.latex(r'F(x) = \begin{cases} 0 & \text{for } x < a \\ \frac{x-a}{b-a} & \text{for } a \leq x \leq b \\ 1 & \text{for } x > b \end{cases}')
+        
+        st.write("""
+        Common applications:
+        - **Random number generation**
+        - **Prior distribution in Bayesian inference**
+        - **Modeling random errors**
+        - **Simple random sampling**
+        
+        Important links:
+        - [**Properties**](https://en.wikipedia.org/wiki/Continuous_uniform_distribution#Properties)
+        - [**Applications**](https://en.wikipedia.org/wiki/Continuous_uniform_distribution#Applications)
+        - [**Relationship to other distributions**](https://en.wikipedia.org/wiki/Continuous_uniform_distribution#Related_distributions)
+        """)

distributions/discrete/__init__.py

@@ -1,4 +1,5 @@
 from .binomial import BinomialDistribution
 from .poisson import PoissonDistribution
 from .multinomial import MultinomialDistribution
-__all__ = ['BinomialDistribution', 'PoissonDistribution', 'MultinomialDistribution']
+from .geometric import GeometricDistribution
+__all__ = ['BinomialDistribution', 'PoissonDistribution', 'MultinomialDistribution', 'GeometricDistribution']

distributions/discrete/binomial.py

@@ -19,11 +19,34 @@ def plot(self, ax):
     def get_formula(self):
         return r'P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}'
 
-    def get_properties(self):
-        return """
-        - Models number of successes in n independent trials
-        - Each trial has probability p of success
-        - Mean = np
-        - Variance = np(1-p)
-        - Support is k ∈ {0,1,...,n}
-        """
+    def get_properties(self, st):
+        st.write("""
+        The **Binomial Distribution** models the number of successes in a fixed number of independent trials.
+        It is characterized by two parameters:
+        - **n (number of trials)**: total number of independent experiments
+        - **p (probability of success)**: probability of success on each trial
+        
+        Key Properties:
+        - **Support**: k ∈ {0,1,...,n}
+        - **Mean**: np
+        - **Variance**: np(1-p)
+        - **Independent trials** (no memory between trials)
+        - **Fixed probability** of success for each trial
+        
+        The probability mass function (PMF) is given by:
+        """)
+        
+        st.latex(r'P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}')
+        
+        st.write("""
+        Common applications:
+        - **Quality control** (number of defective items)
+        - **Clinical trials** (number of successful treatments)
+        - **Polling and surveys** (number of positive responses)
+        - **Genetics** (inheritance patterns)
+        
+        Important links:
+        - [**Properties**](https://en.wikipedia.org/wiki/Binomial_distribution#Properties)
+        - [**Applications**](https://en.wikipedia.org/wiki/Binomial_distribution#Applications)
+        - [**Relationship to other distributions**](https://en.wikipedia.org/wiki/Binomial_distribution#Related_distributions)
+        """)

distributions/discrete/geometric.py

@@ -0,0 +1,50 @@
+import numpy as np
+from scipy.stats import geom
+import streamlit as st
+from ..base import Distribution
+
+class GeometricDistribution(Distribution):
+    def get_parameters(self):
+        self.p = st.slider('Probability of success (p)', 0.01, 1.0, 0.5, 0.01)
+        return {'p': self.p}
+
+    def plot(self, ax):
+        x = np.arange(1, min(20, int(5/self.p)))
+        y = geom.pmf(x, self.p)
+        ax.bar(x, y, alpha=0.8)
+        ax.set_xlabel('Number of Trials Until Success')
+        ax.set_ylabel('Probability')
+
+    def get_formula(self):
+        return r'P(X=k) = (1-p)^{k-1}p'
+
+    def get_properties(self, st):
+        st.write("""
+        The **Geometric Distribution** models the number of trials needed to get the first success in a sequence of independent Bernoulli trials.
+        It is characterized by one parameter:
+        - **p (probability of success)**: probability of success on each trial
+        
+        Key Properties:
+        - **Support**: k ∈ {1,2,3,...}
+        - **Mean = 1/p**
+        - **Variance = (1-p)/p²**
+        - **Memoryless property**: P(X > s + t | X > s) = P(X > t)
+        - **Independent trials** with fixed probability
+        
+        The probability mass function (PMF) is given by:
+        """)
+        
+        st.latex(r'P(X=k) = (1-p)^{k-1}p')
+        
+        st.write("""
+        Common applications:
+        - **Quality control** (number of items inspected until finding a defect)
+        - **Marketing** (number of attempts until first sale)
+        - **Reliability** (number of trials until first failure)
+        - **Game theory** (number of attempts until first win)
+        
+        Important links:
+        - [**Properties**](https://en.wikipedia.org/wiki/Geometric_distribution#Properties)
+        - [**Applications**](https://en.wikipedia.org/wiki/Geometric_distribution#Applications)
+        - [**Relationship to other distributions**](https://en.wikipedia.org/wiki/Geometric_distribution#Related_distributions)
+        """)