Formulas#
Discrete Distributions#
Discrete Uniform \((1, N)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
\frac{1}{N} & \text{if } x = 1, 2, \dots, N \\
0 & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \frac{N+1}{2} \]
\[ Var(X) = \frac{N^2-1}{12} \]
Discrete Uniform \((N_0, N_1)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
\frac{1}{N_1 - N_0 + 1} & \text{if } x = N_0, N_0 + 1, \dots, N_1 - 1, N_1 \\
0 & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \frac{N_1 + N_0}{2} \]
\[ Var(X) = \frac{(N_1 - N_0)(N_1 - N_0 + 2)}{12} \]
Hypergeometric \((N, M, n)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
\frac{\binom{M}{x} \binom{N-M}{n-x}}{\binom{N}{n}} & \text{if } x=0,1,\dots,n \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \frac{nM}{N} \]
\[ Var(X) = \frac{nM}{N}\left(1 - \frac{M}{N}\right)\left(\frac{N-n}{N-1}\right) \]
Bernoulli \((p)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
p, & \text{if } x = 1 \\
1-p, & \text{if } x = 0 \\
0, & \text{otherwise}
\end{cases}
=\begin{cases}
p^x(1-p)^{1-x}, & \text{if } x = 0, 1 \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ EX = p \]
\[ Var(X) = p(1-P) \]
Binomial \((n, p)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
\binom{n}{x}p^x(1-p)^{n-x}, & \text{if } x=0,1,\dots,n \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = np \]
\[ Var(X) = np(1-p) \]
Geometric \((p)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
p(1-p)^{x-1}, & \text{if } x=1,2,\dots \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \frac{1}{p} \]
\[ Var(X) = \frac{1-p}{p^2} \]
Negative Binomial \((r,p)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
\binom{x-1}{r-1}p^r(1-p)^{x-r}, & \text{if } x=r, r+1, r+2, \dots \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ X = W_1 + W_2 + \dots + W_r = \sum_{i=1}^rW_i \]
\[ W_1, W_2, \dots, W_r \sim \text{Geometric}(p) \]
\[ EX = \frac{r}{p} \]
\[ Var(X) = \frac{r(1-p)}{p^2} \]
Poisson \((\lambda)\)#
\[\begin{split}
p_X(x) =
\begin{cases}
\frac{e^{-\lambda} \lambda^x}{x!}, & \text{if } x = 0, 1, 2, \dots \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \lambda \]
\[ Var(X) = \lambda \]
Continuous Distributions#
Continuous Uniform \((a, b)\)#
\[\begin{split}
f_X(x) =
\begin{cases}
\frac{1}{b-a}, & \text{if } a < x < b \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \frac{a+b}{2} \]
\[ Var(X) = \frac{(b-a)^2}{12} \]
Gamma \((\alpha, \beta)\)#
\[\begin{split}
f_X(x) =
\begin{cases}
\frac{1}{\Gamma(\alpha) \beta^\alpha} x^{\alpha-1} e^{-\frac{x}{\beta}} , & \text{if } x>0 \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \alpha \beta \]
\[ Var(X) = \alpha \beta^2 \]
Exponential \((\beta)\)#
\[\begin{split}
f_X(x) =
\begin{cases}
\frac{1}{\beta^\alpha} e^{-\frac{x}{\beta}} , & \text{if } x>0 \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ X \sim \text{Gamma}(1, \beta) \]
\[ E(X) = \beta \]
\[ Var(X) = \beta^2 \]
Chi-Squared \((p)\)#
\[ X \sim \text{Gamma}(\frac{p}{2}, 2) \]
Beta \((\alpha, \beta)\)#
\[\begin{split}
f_X(x) =
\begin{cases}
\frac{1}{B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1}, & \text{if } x \in (0,1) \\
0, & \text{otherwise}
\end{cases}
\end{split}\]
\[ E(X) = \frac{\alpha}{\alpha + \beta} \]
\[ Var(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \]
Normal \((\mu, \sigma)\)#
\[ f_X(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2} \text{ if } x \in \mathbb{R} \]
Joint Distributions#
Marginal Distributions#
\[ p_X(x) = \sum_y p_{X,Y}(x,y) \]
\[ p_Y(y) = \sum_x p_{X,Y}(x,y) \]
\[ f_X(x) = \int_{-\infty}^\infty f_{X,Y}(x,y) \ dy \]
\[ f_Y(y) = \int_{-\infty}^\infty f_{X,Y}(x,y) \ dx \]
Expectations#
\[ Eh(X,Y) = \sum_{(x,y)} h(x,y) p_{X,Y}(x,y) \]
\[ Eh(X,Y) = \iint_{\mathbb{R}^2} h(x,y) f_{X,Y}(x,y) dydx \]
Other#
Sum of Arithmetic Series#
\[ \sum_{x=1}^N x = \frac{N(N+1)}{2} \]
Sum of Geometric Series#
\[ \sum_{n=0}^\infty ar^n = \frac{a}{1-r} \]
Pascal’s Rule#
\[ \binom{n}{r} = \frac{n}{r} \binom{n-1}{r-1} \]
Gamma Function#
\[ \Gamma(\alpha) = \int_0^\infty t^{\alpha - 1} e^{-t} dt, \ \ \alpha > 0 \]
\[ \Gamma(\alpha + 1) = \alpha \Gamma({\alpha}) \]
\[ \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \]
Beta Function#
\[ B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} dx \]