PMF: \begin{equation} \nonumber P_X(k)=\left\{ \begin{array}{l l} p& \quad \text{for } k=1\\ 1-p & \quad \text{for } k=0 \end{array} \right. \end{equation} CDF: \begin{equation} \nonumber F_X(x)=\left\{ \begin{array}{l l} 0 & \quad \text{for } x \lt 0\\ 1-p & \quad \text{for } 0 \leq x \lt 1\\ 1 & \quad \text{for} 1 \leq x \end{array} \right. \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=1-p+pe^s \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=1-p+pe^{i\omega} \end{equation} Expected Value: \begin{equation} \nonumber EX=p \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=p(1-p) \end{equation}

PMF: \begin{equation} \nonumber P_X(k)={n \choose k}p^k(1-p)^{n-k} \quad \text{for } k=0,1,2,\cdots,n \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=(1-p+pe^s)^n \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=(1-p+pe^{i \omega})^n \end{equation} Expected Value: \begin{equation} \nonumber EX=np \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=np(1-p) \end{equation} MATLAB: R = binornd($n$,$p$)

PMF: \begin{equation} \nonumber P_X(k)=p(1-p)^{k-1} \quad \text{for } k=1,2,3,... \end{equation} CDF: \begin{equation} \nonumber F_X(x)=1-(1-p)^{\lfloor x \rfloor} \quad \text{for }x \geq 0 \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=\frac{pe^s}{1-(1-p)e^s} \quad \text{for }s \lt - \ln(1-p) \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=\frac{pe^{i \omega}}{1-(1-p)e^{i\omega}} \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac{1}{p} \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{1-p}{p^2} \end{equation} MATLAB: R = geornd($p$)+1

PMF: \begin{equation} \nonumber P_X(k) = {k-1 \choose m-1} p^{m}(1-p)^{k-m} \quad \text{for } k=m,m+1,m+2,m+3,... \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=\left(\frac{pe^s}{1-(1-p)e^s}\right)^m \quad \textrm{for} \quad s \lt -\log (1-p) \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=\left(\frac{pe^{i \omega}}{1-(1-p)e^{i \omega}}\right)^m \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac{m}{p} \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{m(1-p)}{p^2} \end{equation} MATLAB: R = nbinrnd($m$,$p$)+1

PMF: \begin{equation} \nonumber P_X(x) = \frac{{b \choose x} {r \choose k-x}}{{b+r \choose k}} \quad \text{for }x=\max(0,k-r), \max(0,k-r)+1,..., \min(k,b) \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac{kb}{b+r} \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{kbr}{(b+r)^2} \frac{b+r-k}{b+r-1} \end{equation} MATLAB: R = hygernd($b+r$,$b$,$k$)

PMF: \begin{equation} \nonumber P_X(k) = \frac{e^{-\lambda} \lambda^k}{k!} \quad \text{for } k=0,1,2,\cdots \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=e^{\lambda\left(e^s-1\right)} \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=e^{\lambda\left(e^{i\omega}-1\right)} \end{equation} Expected Value: \begin{equation} \nonumber EX=\lambda \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\lambda \end{equation} MATLAB: R = poissrnd($\lambda$)

PDF: \begin{equation} \nonumber f_X(x)=\lambda e^{-\lambda x}, \quad x>0 \end{equation} CDF: \begin{equation} \nonumber F_X(x)=1- e^{-\lambda x}, \quad x>0 \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=\left(1-\frac{s}{\lambda}\right)^{-1} \quad \textrm{for} \quad s \lt \lambda \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=\left(1-\frac{i\omega}{\lambda}\right)^{-1} \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac1{\lambda} \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac1{\lambda^2} \end{equation} MATLAB: R = exprnd($\mu$), where $\mu=\frac{1}{\lambda}$.

PDF: \begin{align}%\label{} f_X(x) &= \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right) = \left\{\begin{matrix} \frac{1}{2b} \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x \lt \mu \\[8pt] \frac{1}{2b} \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu \end{matrix}\right. \end{align} CDF: \begin{equation} F_X(x)=\left\{\begin{matrix} \frac{1}{2} \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x \lt \mu \\[8pt] 1-\frac{1}{2} \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu \end{matrix}\right. \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=\frac{e^{\mu s}}{1-b^2 s^2} \quad \textrm{for} \quad |s| \lt \frac1{b} \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=\frac{e^{\mu i \omega}}{1+b^2 \omega^2} \end{equation} Expected Value: \begin{equation} \nonumber EX=\mu \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=2b^2 \end{equation}

PDF: \begin{equation}%\label{} f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \end{equation} CDF: \begin{equation} F_X(x)=\Phi\left(\frac{x-\mu}{\sigma}\right) \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=e^{\mu s+\frac1{2}\sigma^2s^2} \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=e^{i \mu \omega-\frac1{2}\sigma^2\omega^2} \end{equation} Expected Value: \begin{equation} \nonumber EX=\mu \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\sigma^2 \end{equation} MATLAB: Z = randn, R = normrnd($\mu$,$\sigma$)

PDF: \begin{equation}%\label{} f_X(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{(a-1)}(1-x)^{(b-1)}, \; \textrm{ for }0 \leq x \leq 1 \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=1+\sum_{k=1}^{\infty}\left(\prod_{r=0}^{k-1} \frac{a+r}{a+b+r}\right)\frac{s^k}{k!} \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac{a}{a+b} \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{ab}{(a+b)^2(a+b+1)} \end{equation} MATLAB: R = betarnd($a$,$b$)

Note: \begin{equation*} \chi^2(n) =Gamma\left(\frac{n}{2},\frac{1}{2}\right) \end{equation*} PDF: \begin{equation} \nonumber f_X(x) = \frac{1}{2^{\frac{n}{2}} \Gamma\left(\frac{n}{2}\right)} x^{\frac{n}{2}-1} e^{-\frac{x}{2}}, \quad \textrm{for } x>0. \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=(1-2s)^{-\frac{n}{2}} \quad \textrm{for} \quad s \lt \frac1{2} \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=(1-2i\omega)^{-\frac{n}{2}} \end{equation} Expected Value: \begin{equation} \nonumber EX=n \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=2n \end{equation} MATLAB: R = chi2rnd($n$)

PDF: \begin{equation} \nonumber f_X(x)=\frac{\Gamma(\frac{n+1}{2})}{\sqrt{n\pi}\Gamma\left(\frac{n}{2}\right)}\left(1+\frac{x^2}{n}\right)^{-\frac{n+1}2} \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber \textrm{undefined} \end{equation} Expected Value: \begin{equation} \nonumber EX=0 \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{n}{n-2} \quad \textrm{for} \quad n>2, \quad \infty \quad \textrm{for } 1 \lt n\leq 2, \quad \textrm{undefined} \quad \textrm{otherwise} \end{equation} MATLAB: R = trnd($n$)

PDF: \begin{equation} \nonumber f_X(x) = \frac{\lambda^{\alpha} x^{\alpha-1} e^{-\lambda x}}{\Gamma(\alpha)}, \quad x>0 \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=\left(1-\frac{s}{\lambda}\right)^{-\alpha} \quad \textrm{for} \quad s \lt \lambda \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac{\alpha}{\lambda} \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{\alpha}{\lambda^2} \end{equation} MATLAB: R = gamrnd($\alpha$,$\lambda$)

PDF: \begin{equation} \nonumber f_X(x) = \frac{\lambda^{k} x^{k-1} e^{-\lambda x}}{(k-1)!}, \quad x>0 \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=\left(1-\frac{s}{\lambda}\right)^{-k} \quad \textrm{for} \quad s \lt \lambda \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac{k}{\lambda} \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{k}{\lambda^2} \end{equation}

PDF: \begin{equation} \nonumber f_X(x)= \frac1{b-a}, \quad x \in [a,b] \end{equation} CDF: \begin{equation} \nonumber F_X(x)=\left\{ \begin{array}{l l} 0& \quad x \lt a\\ \frac{x-a}{b-a} & \quad x \in [a,b)\\ 1 & \quad \textrm{for } x\geq b \end{array} \right. \end{equation} Moment Generating Function (MGF): \begin{equation} \nonumber M_{X}(s)=\left\{ \begin{array}{l l} \frac{e^{sb}-e^{sa}}{s(b-a)} & \quad s\neq0\\ 1 & \quad s=0 \end{array} \right. \end{equation} Characteristic Function: \begin{equation} \nonumber \phi_{X}(\omega)=\frac{e^{i\omega b}-e^{i \omega a}}{i\omega(b-a)} \end{equation} Expected Value: \begin{equation} \nonumber EX=\frac{1}{2}(a+b) \end{equation} Variance: \begin{equation} \nonumber \textrm{Var}(X)=\frac{1}{12}(b-a)^2 \end{equation} MATLAB: U = rand or R = unifrnd($a$,$b$)