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  • Special Distributions

Beta Distribution

• A random variable $X$ has a beta distribution
• Standardized between $0,1$
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  • STA260 Lecture 14
• The Beta distribution is a continuous probability distribution defined on the interval $(0,1)$, parameterized by two positive shape parameters, $\alpha$ and $\beta$.
• The probability density function (PDF) of a Beta distribution is defined as $\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}{x}^{\alpha -1}(1-x{)}^{\beta -1}$ for $0<x<1$.
• A Beta distribution can be derived through the mathematical transformation of other standard continuous variables.
  • If a random variable $X$ is uniformly distributed such that $X\sim U(0,1)$, the transformation $Y=X^{\frac{1}{\alpha }}$ results in the distribution $Y\sim \mathrm{Beta}(\alpha ,1)$.