Probability Distributions

Exponential

\[Y\sim\text{Exp}(\mu)=\frac{1}{\mu}e^{-\frac{y}{\mu}}\]

\[Y\sim\text{Exp}(\lambda)=\lambda e^{-\lambda y}\]

Normal

\[Y\sim\text{N}(\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{y-\mu}{\sigma})}\]

Poisson

\[Y\sim\text{Pois}(\lambda)=\begin{cases}\frac{\lambda^y}{y!}e^{-\lambda} & y\geq 0 \\ 0 & \text{Otherwise}\end{cases}\]

Uniform

\[Y\sim U(a,b)=\frac{1}{b-a}\]

Geometric

\[Y\sim Geom(p) = \begin{cases}p(1-p)^y & y\geq 0\\0& \text{Otherwise}\end{cases}\]

Gamma

\[Y\sim \text{Gamma}(\alpha,\beta)=\begin{cases}\frac{y^{\alpha-1}e^{\frac{-y}{\beta}}}{\beta^\alpha \Gamma(\alpha)}&y\geq 0\\0& \text{Otherwise}\end{cases}\]

Note:

• Standard Gamma Distribution has \(\beta = 1\)

• \(\Gamma(\alpha =1)=1\)

• \(\Gamma(\alpha =2)=1\)

• \(\Gamma(\alpha =3)=2\)

• \(\Gamma(\alpha =4)=6\)

Chi-squared

\[Y\sim \chi^2(k)=\begin{cases}\frac{y^{\frac{1}{2}k-1}e^{-\frac{1}{2}y}}{2^{\frac{1}{2}k}\Gamma (\frac{1}{2}k)}&y>0\\0&\text{Otherwise}\end{cases}\]