Probability Distribution
Elementary
In the realm of Mathematical Statistics, an unavoidable topic for statisticians is Probability Distributions, which is covered here.
Basic Distributions
- Uniform Distribution $U(a,b)$
- Binomial Distribution $\text{Bin} (n,p)$
- Geometric Distribution $\text{Geo} (p)$
- Negative Binomial Distribution $\text{NB} (r,p)$
- Poisson Distribution $\text{Poi} ( \lambda )$
- Exponential Distribution $\exp ( \lambda)$
- Normal Distribution $N \left( \mu,\sigma^{2} \right)$
Sampling Distributions
- Gamma Distribution $\Gamma (k,\theta)$
- Beta Distribution $\text{Beta} (\alpha,\beta)$
- Chi-Square Distribution $\chi^{2}(r)$
- F-Distribution $F \left( r_{1} , r_{2} \right)$
- t-Distribution $t \left( \nu \right)$
- Cauchy Distribution: A Distribution without a Mean $C$
Advanced
In the actual world of formal sciences, Probability Distribution Theory extends beyond the scope of statistics and finds applications in numerous areas. However, well-organized references akin to textbooks are scarce. The reality is that one often needs to explore various papers and study each distribution separately, but in the future, this blog aims to comprehensively organize this information.
Special Distributions
- Pareto Distribution
- Log-Normal Distribution
- Skew-Normal Distribution
- Weibull Distribution
- Non-central Chi-Square Distribution
- Non-central F-Distribution
Multivariate Distributions
- Multinomial Distribution $M_{k} \left( n, \mathbf{p} \right)$
- Multivariate Normal Distribution $N_{p} \left( \mu , \Sigma \right)$
- Multivariate t-Distribution $t_{p} \left(\nu; \mu , \Sigma \right)$
Directional Distributions
- Reason Modified Bessel Function of the First Kind $I_{\nu}$ Appears in Directional Statistics
- Von Mises Distribution $\text{vM} \left( \mu , \kappa \right)$
- Bivariate Von Mises Distribution $\text{vM}^{2} \left( \mu , \nu , \kappa_{1} , \kappa_{2} \right)$
- Von Mises-Fisher Distribution $\text{vMF}_{p} \left( \mu , \kappa \right)$
- Bingham-Mardia Distribution
- Kent Distribution
References
- Casella. (2001). Statistical Inference(2nd Edition)
- Hogg et al. (2013). Introduction to Mathematical Statistcs(7th Edition)
All posts
- Expectation of the Power of Normally Distributed Random Variables with Mean Zero
- Binomial Distribution
- Mean and Variance of the Bernoulli Distribution
- Binomial Distribution
- Mean and Variance of the Binomial Distribution
- Geometric Distribution
- Mean and Variance of the Geometric Distribution
- Differences Between the Two Definitions of the Geometric Distribution
- Negative Binomial Distribution
- Mean and Variance of the Negative Binomial Distribution
- Poisson Distribution
- Mean and Variance of the Poisson Distribution
- Exponential Distribution
- Mean and Variance of Exponential Distribution
- The Relationship between Exponential Distribution and Poisson Distribution
- Exponential Distribution's Memorylessness
- Geometric Distribution's Memorylessness
- Gamma Distribution
- Mean and Variance of Gamma Distribution
- The Relationship between Gamma Distribution and Poisson Distribution
- Relationship between Gamma Distribution and Exponential Distribution
- The Relationship Between the Gamma Distribution and the Chi-Squared Distribution
- Beta Distribution
- Mean and Variance of the Beta Distribution
- Derivation of the Beta Distribution from Two Independent Gamma Distributions
- Chi-Squared Distribution
- The Mean and Variance of the Chi-Squared Distribution
- F-distribution
- Mean and Variance of the F-distribution
- Derivation of F-distribution from Two Independent Chi-squared Distributions
- Normal Distribution
- Normal Distribution: Mean and Variance
- The Square of a Standard Normal Distribution Follows a Chi-Square Distribution with One Degree of Freedom
- Derivation of the Student's t-Distribution from Independent Normal Distributions and the Chi-Squared Distribution
- t-Distribution
- Mean and Variance of the t-Distribution
- Cauchy Distribution: A Distribution Without a Mean
- The Poisson Distribution as a Limiting Distribution of the Binomial Distribution
- Derivation of the Standard Normal Distribution as a Limiting Distribution of the Binomial Distribution
- Derivation of the Standard Normal Distribution as the Limiting Distribution of the Poisson Distribution
- Deriving Standard Normal Distribution as a Limiting Distribution of Student's t-Distribution
- Multivariate Normal Distribution
- Multivariate t-Distribution
- Approximation of Normal Distribution Variance Stabilization from a Binomial Distribution
- Log-Normal Distribution
- Pareto Distribution
- Rayleigh Distribution
- Weibull Distribution
- Derivation of Beta Distribution from F-Distribution
- Derivation of the F-distribution from the t-distribution
- Sufficient Statistics and Maximum Likelihood Estimators for the Binomial Distribution
- Sufficient Statistics and Maximum Likelihood Estimators for the Geometric Distribution
- Sufficient Statistics and Maximum Likelihood Estimators of the Poisson Distribution
- Sufficient Statistics and Maximum Likelihood Estimators of Exponential Distributions
- Sufficient Statistics and Maximum Likelihood Estimators of a Normal Distribution
- Sufficient Statistics of the Gamma Distribution
- Sufficient Statistics for the Beta Distribution
- Chi-Square Distribution's Sufficient Statistics
- Proof of Normality of Regression Coefficients
- Reasons Why the Modified Bessel Function of the First Kind Appears in Directional Statistics
- Polynomial Distribution
- Entropy of Normal Distribution
- Derivation of the Covariance Matrix of the Multinomial Distribution
- Proof of Pearson's Theorem
- Von Mises Distribution
- Bivariate von Mises Distribution
- Von Mises-Fisher Distribution
- Why Normal Distribution
- Bingham-Mardia Distribution
- Kent Distribution
- Noncentral Chi-Squared Distribution
- Nonsymmetric F-distribution
- Linear Transformations of Multivariate Normal Distributions
- Independence and Zero Correlation are Equivalent in Multivariate Normal Distribution
- Conditional Mean and Variance of the Multivariate Normal Distribution