Common Probability Distributions

$$\text{PMF: } P(X=x)=p(1-p)^{x-1}; \\ \quad \\ \text{for } x \in \mathbb{Z}_{> 0}$$
$$\text{PMF: } P(X=x)={n \choose x}p^x(1-p)^{n-x}; \\ \quad \\ \text{for } x=0,1,\dots,n$$
$$\text{PMF: } P(X=x)=\frac{\lambda^xe^{-\lambda}}{x!}; \\ \quad \\ \text{for } x \in \mathbb{Z}_{\geq 0}$$
$$\text{PMF: } P(X=x)={{x-1} \choose {r-1}}p^r(1-p)^{x-r}; \\ \quad \\ \text{for } x=r,r+1,\dots \\ \text{where } r \in \mathbb{Z}_{>0}$$
$$\text{PMF: } P(X=x)=\frac{{r \choose x}{{N-r} \choose {n-x}}}{{N \choose n}}; \\ \quad \\ \text{for } x=0,1,\dots,n \quad \text{if } n \leq r \\ \text{or } x=0,1,\dots,r \quad \text{if } n > r \\ \text{where } N, n, r \in \mathbb{Z}_{>0}$$
$$\text{PDF: } f(x)=\frac{1}{\theta_2-\theta_1}; \quad \text{for } \theta_1 \leq x \leq \theta_2 \\ \text{where } \theta_1, \theta_2 \in \mathbb{R}$$
$$\text{PDF: } f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\left[\frac{-(x - \mu)^2}{2\sigma^2}\right]}; \text{for } x \in \mathbb{R}$$
$$\text{PDF: } f(x)=\frac{1}{\beta}e^{-\frac{x}{\beta}}; \quad \text{for } x, \beta \in \mathbb{R}_{>0}$$
$$\text{PDF: } f(x)=\left[\frac{1}{\Gamma(\alpha)\beta^\alpha}\right]x^{\alpha-1}e^{-\frac{x}{\beta}}; \\ \quad \\ \text{for } x, \alpha, \beta \in \mathbb{R}_{>0} \\ \quad \\ \text{where } \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt \quad \text{for } z \in \mathbb{R}_{>0} \\ \text{and } \Gamma(n)=\left( n-1 \right)! \quad \text{for } n \in \mathbb{Z}_{>0}$$
$$\text{PDF: } f(x)=\frac{x^{\left( \nu/2-1 \right)}e^{-x/2}}{2^{\nu/2}\Gamma \left( \nu/2 \right)}; \\ \quad \\ \text{for } x \in \mathbb{R}_{>0} \text{ if } \nu=1 \\ \text{and } x \in \mathbb{R}_{\geq 0} \text{ otherwise} \\ \text{where } \nu \in \mathbb{Z}_{>0} $$
$$\text{PDF: } f(x)=\left[\frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} \right]x^{\alpha-1}(1-x)^{\beta-1}; \\ \quad \\ \text{for } 0 < x, \alpha, \beta < \infty \\ \quad \\ \text{where } \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt \quad \text{for } z \in \mathbb{R}_{>0} \\ \text{and } \Gamma(n)=\left( n-1 \right)! \quad \text{for } n \in \mathbb{Z}_{>0}$$
$$\text{PDF: } f(x) = \\ \quad \\ \sum_{k=0}^{\infty}\frac{e^{-\frac{\delta}{2}}\left(\frac{\delta}{2} \right)^k\left( \frac{\nu_1x}{\nu_2} \right)^{\frac{\nu_1}{2}+k} \left( \frac{\nu_2}{\nu_2+\nu_1x} \right)^{\frac{\nu_1+\nu_2}{2}+k}}{\text{B}\left(\frac{\nu_2}{2}, \frac{\nu_1}{2}+k \right)k!x}; \\ \quad \\ \text{for } \nu_1, \nu_2 \in \mathbb{Z}_{>0} \text{, } \delta \in \mathbb{R}_{\geq 0} \\ \quad \\ \text{ and } x \in \mathbb{R}_{> 0} \text{ if } \nu_1=1 \text{, else } x \in \mathbb{R}_{\geq 0} \\ \quad \\ \text{where } \text{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}dt$$
$$\text{PDF: } f(x)=\frac{\Gamma \left( \frac{\nu + 1}{2} \right)}{\Gamma \left( \frac{\nu}{2} \right) \sqrt{\nu\pi}} \left( 1 + \frac{x^2}{\nu}\right)^{-\frac{\nu + 1}{2}}; \\ \quad \\ \text{for } \nu \in \mathbb{Z}_{>0} \\ \quad \\ \text{where } \Gamma(\nu)=\left( \nu-1 \right)!$$

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