Distributions

Main references: [1].

Notations

• $Y$: a random variable
• $\text{pr}\text{rdb}Y\mid \theta$: the probability of $Y$
• $p(y\mid \theta )=\text{pr}\text{rdb}Y=y\mid \theta$: the discrete probability density function
• $f(y\mid \theta )={\displaystyle \frac{\text{dl}1}{\text{dl}1y}}\text{pr}\text{rdb}Y\le y\mid \theta$: the continuous probability density function
• $\text{Exp}\text{rdb}Y$: the expectation of $Y$
• $\text{Var}\text{rdb}Y$: the variance of $Y$

Distributions

• $\text{distbinom}(n,\theta )$: Binomial distribution.
• $\text{distbeta}(\alpha ,\beta )$: Beta distribution.
• $\text{distpois}(\lambda )$: Poisson distribution.
• $\text{distgamma}(\lambda )$: Gamma distribution.
• $\text{distexp}(\lambda )$: Exponential distribution.
• $\text{distnormal}(\mu ,{\sigma }^2)$: Nomral distribution.

pdfs

• ${\displaystyle \text{pdfbinom}(y,n,\theta )={\displaystyle (\genfrac{}{}{0ex}{}{n}{y})}{\theta }^y(1-\theta {)}^{n-y}}$.
• ${\displaystyle \text{pdfbeta}(\theta ,\alpha ,\beta )={\displaystyle \frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}}{\theta }^{\alpha -1}(1-\theta {)}^{\beta -1}}$.
• ${\displaystyle \text{pdfpois}}$
• ${\displaystyle \text{pdfgamma}}$
• ${\displaystyle \text{pdfexp}}$
• ${\displaystyle \text{pdfnormal}}$

[1]

Hoff, P. D. (2009). A first course in bayesian statistical methods. Springer.