Distributions
Main references: [1].
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\[ \newcommand{\distbinom}{\operatorname{B}} \newcommand{\distbeta}{\operatorname{Beta}} \newcommand{\distgamma}{\operatorname{Gamma}} \newcommand{\distexp}{\operatorname{Exp}} \newcommand{\distpois}{\operatorname{Poisson}} \newcommand{\distnormal}{\operatorname{\mathcal N}} \]
Notations
- \(Y\): a random variable
- \(\Pr(Y\mid\theta)\): the probability of \(Y\)
- \(p(y\mid\theta)=\Pr(Y=y\mid\theta)\): the discrete probability density function
- \(f(y\mid\theta)=\dfrac{\dl1}{\dl1y}\Pr(Y\leq y\mid\theta)\): the continuous probability density function
- \(\Exp\qty(Y)\): the expectation of \(Y\)
- \(\Var\qty(Y)\): the variance of \(Y\)
Distributions
- \(\distbinom(n,\theta)\): Binomial distribution.
- \(\distbeta(\alpha,\beta)\): Beta distribution.
- \(\distpois(\lambda)\): Poisson distribution.
- \(\distgamma(\lambda)\): Gamma distribution.
- \(\distexp(\lambda)\): Exponential distribution.
- \(\distnormal(\mu,\sigma^2)\): Nomral distribution.
pdfs
- \(\displaystyle \pdfbinom(y, n, \theta)=\dbinom{n}{y} \theta^{y}(1-\theta)^{n-y}\).
- \(\displaystyle \pdfbeta(\theta, \alpha, \beta)=\dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\theta^{\alpha-1}(1-\theta)^{\beta-1}\).
- \(\displaystyle \pdfpois\)
- \(\displaystyle \pdfgamma\)
- \(\displaystyle \pdfexp\)
- \(\displaystyle \pdfnormal\)