Beta Distribution
Tags: #Math #StatisticsEquation
$$X \sim Beta(\alpha,\beta), \\ f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \\ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, \\ F(x)=I_{x}(\alpha+\beta)$$Latex Code
X \sim Beta(\alpha,\beta), \\
f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \\
B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, \\
F(x)=I_{x}(\alpha+\beta)
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Introduction
Equation
$$X \sim Beta(\alpha,\beta)$$ $$f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$$ $$B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$$ $$F(x)=I_{x}(\alpha+\beta)$$
Latex Code
X \sim Beta(\alpha,\beta), \\
f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \\
B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, \\
F(x)=I_{x}(\alpha+\beta)
Explanation
Latex code for the Beta Distribution.
- Shape parameter: $$\alpha, \beta$$
- PDF for Beta Distribution: $$f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$$
- Beta Function: $$B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$$
- CDF for Beta Distribution, which is the regularized incomplete beta function: $$F(x)=I_{x}(\alpha+\beta)$$
- Mean for Beta Distribution: $$E[X] = \frac{\alpha}{\alpha+\beta}$$
- Variance for Beta Distribution: $$var[X] = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} $$
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