Equation Database

EQUATION LIST

math

Arithmetic and Geometric Progressions

#math #arithmetic #geometric #progressions

$$S_{n}=a+(a+d)+(a+2d)+...+[a+(n-1)d]=\frac{n}{2}[2a+(n-1)d] \\ S_{n}=a+ar+ar^{2}+...+ar^{n-1}=a\frac{1-r^{n}}{1-r}$$

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Bessel Equation

#math #bessel equation

$$x^{2}\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}+x\frac{\mathrm{d} y}{\mathrm{d} x}+(x^{2}-m^{2})y=0 \\ J_{m}(x)=\sum^{\infty}_{k=0}\frac{(-1)^{k} (x/2)^{m+2k}}{k!(m+k)!}$$

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Binomial Expansion

#math #binomial #expansion

$$(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+...+ C^{r}_{n}x^{r} +...+x^{n} \\ C^{r}_{n}=\frac{n!}{r!(n-r)!}$$

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Convergence of Series

#math #convergence #series

$$S_{n}=u_{1}+u_{2}+...+u_{n} \\ \text{Converge AS } n \rightarrow \infty \text{, If} \lim_{n \rightarrow \infty} |\frac{u_{n+1}}{u_{n}}| < 1$$

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Determinants of a Matrix

#math #determinants

$$|A|=\sum_{i,j,k,...}\epsilon_{ijk}A_{1i}A_{2j}A_{3k}...$$

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Diffusion Conduction Equation

#math #diffusion #conduction

$$\kappa u_{xx} = u_{t} \\ \frac{\partial \Psi}{\partial t}=\kappa \triangledown^{2} \Psi$$

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Eigenvalues and Eigenvectors

#math #eigenvalues #eigenvectors

$$A\mathbf{u}=\lambda\mathbf{u} \\ P_{n}(\lambda)=|A-\lambda I|$$

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Fourier Series

#math #fourier series

$$y(x)=c_{0}+\sum^{M}_{m=1}c_{m}\cos mx+\sum^{M^{'}}_{m=1}s_{m}\sin mx \\ c_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}y(x) \mathrm{d} x \\ c_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \cos mx \mathrm{d} x \\ s_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \sin mx \mathrm{d} x$$

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Fourier Transforms

#math #fourier transforms

$$y(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{y}(\omega) e^{i\omega t} \mathrm{d} \omega \\ \hat{y}(\omega)=\int_{-\infty}^{\infty} y(t) e^{-i\omega t} \mathrm{d} t$$

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Heat Equation

#math #heat equation

$$u_{t}={\alpha}^{2}u_{xx}$$

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Laplace Equation

#math #laplace equation

$$u_{xx} = 0$$

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Laplace Transform

#math #laplace transform

$$\bar{y}(s)=\mathcal{L}(y(t))=\int^{\infty}_{0} e^{-st} y(t) \mathrm{d}t$$

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Legendre Equation

#math #legendre equation

$$(1-x^{2})\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}-2x\frac{\mathrm{d} y}{\mathrm{d} x}+l(l+1)y=0 \\ P_{l}(x)=\frac{1}{2^{l}l!}(\frac{\mathrm{d}}{\mathrm{d} x})^{l}(x^2-1)^{l}\\ P_{l}(x)=\frac{1}{l}[(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}(x)]$$

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Maclaurin Series

#math #maclaurin series

$$y(x)=y(a+u)=y(0)+x\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{1}{2!}x^{2}\frac{\mathrm{d}^2 y}{\mathrm{d} x^{2}}+\frac{1}{3!}x^{3}\frac{\mathrm{d}^3 y}{\mathrm{d} x^{3}}+...$$

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Plane Wave Expansion

#math #plane wave expansion

$$\exp(ikz)=\exp(ikr\cos \theta=\sum^{\infty}_{l=0}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos \theta)$$

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Power Series with Real Variables

#math #power #series

$$e^{x}=1+x+\frac{x^{2}}{2!}+...+\frac{x^{n}}{n!}+... \\ \ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} + ... + (-1)^{n+1}\frac{x^{n}}{n!} +... \\ \cos(x) = \frac{e^{ix}+e^{-ix}}{2}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+...\\ \sin(x) = \frac{e^{ix}-e^{-ix}}{2i}=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+...$$

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Regression Least Squares Fitting

#math #regression

$$y_{i}=\alpha + \beta (x_{i} - \bar{x}) + \epsilon_{i} \\ \hat{\alpha}=\bar{y} \\ \hat{\beta}=\frac{s^{2}_{xy}}{s^{2}_{x}}$$

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Spherical Harmonics Equation

#math #spherical harmonics

$$[\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}(\sin \theta \frac{\partial}{\partial \theta}) + \frac{1}{\sin^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0 \\ Y^{m}_{l}(\theta,\phi)=\sqrt{\frac{2l+1}{4 \pi} \frac{(l-|m|)!}{(l+|m|)!}}P^{m}_{l}(\cos \theta) e^{im \phi} \times \begin{cases}(-1)^{m} & m\ge 0 \\ 1 & m <0 \end{cases}$$

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Taylor Series

#math #taylor series

$$y(x)=y(a+u)=y(a)+u\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{1}{2!}u^{2}\frac{\mathrm{d}^2 y}{\mathrm{d} x^{2}}+\frac{1}{3!}u^{3}\frac{\mathrm{d}^3 y}{\mathrm{d} x^{3}}+...$$

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Wave Equation

#math #wave equation

$$u_{tt}=c^{2}u_{xx}$$

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Area of a Circle

#Math #Geometry

$$A = \pi r^2$$

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Area of a Ellipse

#Math #Geometry

$$A = \pi r_1 r_2$$

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Area of a Parallelogram

#Math #Geometry

$$A = bh$$

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Area of a Rectangle

#Math #Geometry

$$A = lw$$

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Area of a Regular Polygon

#Math #Geometry

$$A = \frac{{nsr}}{2} = \frac{{pr}}{2}$$

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Area of a Rhombus

#Math #Geometry

$$A = \frac{{x_1 x_2 }}{2}$$

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Area of a Sector

#Math #Geometry

$$A = \frac{{\theta r^2 }}{2}$$

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Area of a Square

#Math #Geometry

$$A = x^2$$

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Area of a Trapezoid

#Math #Geometry

$$A = \frac{1}{2}(x_1 + x_2 )h$$

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Area of a Triangle

#Math #Geometry

$$A = \frac{1}{2}(x_1 + x_2 )h$$

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Binomial Distribution

#Math #Statistics

$$X \sim B(n,p) \\f(x)=\begin{pmatrix}n\\ x\end{pmatrix}p^{x}q^{n-x}=C^{k}_{n}p^{x}q^{n-x},q=1-p\\\text{Binominal Mean}\ \mu=np\\\text{Binominal Variance}\ \sigma^2=npq$$

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Complex Numbers

#math #complex numbers

$$z=x+iy=r(\cos \theta + i \sin \theta)=re^{i(\theta+2n\pi)} \\ z^{*}=x-iy=r(\cos \theta - i \sin \theta)=re^{-i\theta} \\ zz^{*}=|z|^{2}=x^{2}+y^{2}$$

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De Moivre's Theorem

#math #complex number #De Moivre

$$(\cos \theta + i \sin \theta)^{n}=e^{in\theta}=\cos n\theta + i \sin n\theta$$

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Exponential Distribution

#Math #Statistics

$$f(x, \lambda)=\lambda e^{-\lambda x} \\ F(x, \lambda)=1 - e^{-\lambda x}$$

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Fisher Skewness

#Math #Statistics

$$\gamma_1 = \frac{{\mu_3 }}{{\mu_2 ^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }} = \frac{{\mu_3 }}{{\sigma ^3 }}$$

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Gamma Distribution

#Math #Statistics

$$\Gamma \left( a \right) = \int\limits_0^\infty {s^{a - 1} } e^{ - s} ds \\ P(x) = \frac{x^{\alpha-1} e^{-frac{x}{\theta}}}{\Gamma(\alpha) \theta^{\alpha}} \\ \mu = \alpha \theta \\ \sigma^{2} = \alpha \theta^{2} \\ \gamma_{1} = \frac{2}{\sqrt{\alpha}} \\ \gamma_{2} = \frac{6}{\alpha}$$

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Normal Gaussian Distribution

#Math #Statistics

$$X \sim \mathcal{N}(\mu,\sigma^2) \\ f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp{[-\frac{(x-\mu)^{2}}{2\sigma^{2}}]}$$

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Poisson Distribution

#Math #Statistics

$$X \sim \pi(\mu) \\f(x)=\frac{\mu^{x}}{x!}e^{-\mu}\\ \text{Poisson Mean} \mu \\ \text{Poisson Variance}\sigma^2=\mu$$

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Power Series for Complex Variables

#math #complex variables

$$e^{z}=1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+...+\frac{z^{n}}{n!}+...\\ \sin z=z-\frac{z^{3}}{3!}+\frac{z^{5}}{5!}-...\\ \cos z=1-\frac{z^{2}}{2!}+\frac{z^{4}}{4!}-...\\ \ln (1+z)=1-\frac{z^{2}}{2!}+\frac{z^{3}}{3!}-...\\ (1+z)^{n}=1+nz+\frac{n(n-1)}{2!}z^{2}+\frac{n(n-1)(n-2)}{3!}z^{3}+...$$

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gaussian process

#math #gaussian process

$$\log p(y|X) \propto -[y^{T}(K + \sigma^{2}I)^{-1}y+\log|K + \sigma^{2}I|] \\ f(X)=\[f(x_{1}),f(x_{2}),...,f(x_{N}))\]^{T} \sim \mathcal{N}(\mu, K_{X,X}) \\ f_{*}|X_{*},X,y \sim \mathcal{N}(\mathbb{E}(f_{*}),\text{cov}(f_{*})) \\ \text{cov}(f_{*})=K_{X_{*},X_{*}}-K_{X_{*},X}[K_{X,X}+\sigma^{2}I]^{-1}K_{X,X_{*}}$$

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Area of a Equilateral Triangle

#Math #Geometry

$$A = \frac{{h^2 \sqrt 3 }}{3}$$

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Cartesian to Cylindrical Coordinates

#Math #Geometry

$$\begin{array}{*{20}c} {x = r\cos \theta } & {r = \sqrt {x^2 + y^2 } } & {} \\ {y = r\sin \theta } & {\theta = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} & {} \\ {z = z} & {z = z} & {} \\ \end{array}$$

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Cartesian to Polar Coordinates

#Math #Geometry

$$\begin{array}{*{20}c} {x = r\cos \theta } & {r = \sqrt {x^2 + y^2 } } \\ {y = r\sin \theta } & {\theta = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} \\ \end{array}$$

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Cartesian to Spherical Coordinates

#Math #Geometry

$$\begin{array}{*{20}c} {x = R\sin \theta \cos \phi } & {R = \sqrt {x^2 + y^2 + z^2 } } & {} \\ {y = R\sin \theta \sin \phi } & {\phi = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} & {} \\ {z = R\cos \theta } & {\theta = \cos ^{ - 1} \left( {\frac{z}{{\sqrt {x^2 + y^2 + z^2 } }}} \right)} & {} \\ \end{array}$$

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Circle

#Math #Geometry

$$\left( {x - x_0 } \right)^2 + \left( {y - y_0 } \right)^2 = R^2$$

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Cylindrical to Spherical Coordinates

#Math #Geometry

$$\begin{array}{*{20}c} {r = R\sin \theta } & {R = \sqrt {r^2 + z^2 } } & {} \\ {z = R\sin \theta } & {\phi = \theta } & {} \\ {\theta = \phi } & {\theta = \tan ^{ - 1} \left( {\frac{r}{z}} \right)} & {} \\ \end{array}$$

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Distance Between Two Points 2D

#Math #Geometry

$$d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }$$

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Distance Between Two Points 3D

#Math #Geometry

$$d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 + \left( {z_1 - z_2 } \right)^2 }$$

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Eccentricity of a Hyperbola

#Math #Geometry

$$\varepsilon = \frac{{\sqrt {a^2 + b^2 } }}{a}$$

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Eccentricity of an Ellipse

#Math #Geometry

$$\varepsilon = \frac{{\sqrt {a^2 - b^2 } }}{a}$$

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Ellipse

#Math #Geometry

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1 $$

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Hyperbola

#Math #Geometry

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} - \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1 $$

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Hyperbolic Paraboloid

#Math #Geometry

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} - \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)}}{c} $$

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Parabola

#Math #Geometry

$$\left( {y - y_0 } \right)^2 = 4a\left( {x - x_0 } \right)$$

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Plane

#Math #Geometry

$$Ax + By + Cz + D = 0$$

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Sphere

#Math #Geometry

$$\left( {x - x_0 } \right)^2 + \left( {y - y_0 } \right)^2 + \left( {z - z_0 } \right)^2 = R^2 $$

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Surface area of a Cuboid

#Math #Geometry

$$S = 2lw + 2lh + 2wh$$

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Surface area of a Sphere

#Math #Geometry

$$S = 4\pi r^2$$

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Surface area of a cube

#Math #Geometry

$$S = 6x^2$$

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Surface area of a cylinder

#Math #Geometry

$$S = 2\pi r^2 + 2\pi rh$$

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Bernoulli Distribution

#Math #Statistics

$$Pr(X=1) = p = 1- Pr(X=0) = 1 - q, \\ f(x)=p \text{ if } k = 1 \text{ else } q \text{ if } k = 0$$

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Beta Binomial Distribution

#Math #Statistics

$$x \sim f(x | n, \alpha, \beta), \\ f(x | n, \alpha, \beta) = \int_{0}^{1} \text{Bin}(x|n,p)\text{Beta}(p|\alpha,\beta) dp , \\ f(x | n, \alpha, \beta) = C^{x}_{n} \frac{B(x+\alpha, n -x + \beta)}{B(\alpha,\beta)}$$

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Beta Distribution

#Math #Statistics

$$X \sim Beta(\alpha,\beta), \\ f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \\ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, \\ F(x)=I_{x}(\alpha+\beta)$$

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Chi-Squared Distribution

#Math #Statistics

$$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k), \\ f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2}, \\ F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2})$$

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Ellipsoid

#Math #Geometry

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} + \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }} = 1 $$

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Elliptic Cone

#Math #Geometry

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }}$$

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Elliptic Cylinder

#Math #Geometry

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1 $$

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Elliptic Paraboloid

#Math #Geometry

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)}}{c} $$

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Euler identity

#math #euler

$$e^{i\pi} + 1 = 0$$

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Gaussian Integral

#math #calculus

$${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.} $$

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Geometric Distribution

#Math #Statistics

$$Pr(X=k) = (1-p)^{k-1}q, \\ f(x)=(1-p)^{k-1}q, \\ F(x)=1 - (1-p)^{[x]}$$

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Gumbel Distribution

#Math #Statistics

$$x \sim \text{Gumbel}(\mu,\beta), \\ \frac{1}{\beta} e^{-(z + e^{-z})}, z=\frac{x - \mu}{\beta}, \\ e^{-e^{-(x-\mu)/\beta}}$$

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Laplace Distribution

#Math #Statistics

$$x \sim \text{Laplace}(\mu,b), \\ f(x | \mu,b) =\frac{1}{2b} \exp (-\frac{|x-\mu|}{b}), \\ F(x | \mu,b) = \frac{1}{2} \exp (\frac{x - \mu}{b}) \text{ if } x \le \mu, 1 - \frac{1}{2} \exp (-\frac{x - \mu}{b}) \text{ if } x \ge \mu$$

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Poisson Binomial Distribution

#Math #Statistics

$$Pr(K = k) = \sum_{A \in F_{k}} \prod_{i \in A} p_{i} \prod_{j \in A_{c}} (1-p_{j})$$

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Pythagorean Theorem

#math #geometry

$$a^{2}+b^{2}=c^{2}$$

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Riemann Hypothesis

#math

$$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots $$ $$ \zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdots$$

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Spiral of Archimedes

#Math #Geometry

$$r = a \theta$$

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Student t-Distribution

#Math #Statistics

$$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}, \\ F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}, \\ F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}$$

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Uniform Distribution

#Math #Statistics

$$X \sim U(a,b) \\ f(x)=\frac{1}{b-a} \text{for} a \le x \le b \text{else} 0 \\ F(x)=\frac{x-a}{b-a} \text{for} a \le x \le b, 1 \text{for} x > b, 0 \text{for} x < a$$

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statistics

Bernoulli Distribution

#Math #Statistics

$$Pr(X=1) = p = 1- Pr(X=0) = 1 - q, \\ f(x)=p \text{ if } k = 1 \text{ else } q \text{ if } k = 0$$

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Beta Binomial Distribution

#Math #Statistics

$$x \sim f(x | n, \alpha, \beta), \\ f(x | n, \alpha, \beta) = \int_{0}^{1} \text{Bin}(x|n,p)\text{Beta}(p|\alpha,\beta) dp , \\ f(x | n, \alpha, \beta) = C^{x}_{n} \frac{B(x+\alpha, n -x + \beta)}{B(\alpha,\beta)}$$

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Beta Distribution

#Math #Statistics

$$X \sim Beta(\alpha,\beta), \\ f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \\ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, \\ F(x)=I_{x}(\alpha+\beta)$$

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Binomial Distribution

#Math #Statistics

$$X \sim B(n,p) \\f(x)=\begin{pmatrix}n\\ x\end{pmatrix}p^{x}q^{n-x}=C^{k}_{n}p^{x}q^{n-x},q=1-p\\\text{Binominal Mean}\ \mu=np\\\text{Binominal Variance}\ \sigma^2=npq$$

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Chi-Squared Distribution

#Math #Statistics

$$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k), \\ f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2}, \\ F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2})$$

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Exponential Distribution

#Math #Statistics

$$f(x, \lambda)=\lambda e^{-\lambda x} \\ F(x, \lambda)=1 - e^{-\lambda x}$$

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Fisher Skewness

#Math #Statistics

$$\gamma_1 = \frac{{\mu_3 }}{{\mu_2 ^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }} = \frac{{\mu_3 }}{{\sigma ^3 }}$$

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Gamma Distribution

#Math #Statistics

$$\Gamma \left( a \right) = \int\limits_0^\infty {s^{a - 1} } e^{ - s} ds \\ P(x) = \frac{x^{\alpha-1} e^{-frac{x}{\theta}}}{\Gamma(\alpha) \theta^{\alpha}} \\ \mu = \alpha \theta \\ \sigma^{2} = \alpha \theta^{2} \\ \gamma_{1} = \frac{2}{\sqrt{\alpha}} \\ \gamma_{2} = \frac{6}{\alpha}$$

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Geometric Distribution

#Math #Statistics

$$Pr(X=k) = (1-p)^{k-1}q, \\ f(x)=(1-p)^{k-1}q, \\ F(x)=1 - (1-p)^{[x]}$$

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Gumbel Distribution

#Math #Statistics

$$x \sim \text{Gumbel}(\mu,\beta), \\ \frac{1}{\beta} e^{-(z + e^{-z})}, z=\frac{x - \mu}{\beta}, \\ e^{-e^{-(x-\mu)/\beta}}$$

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Laplace Distribution

#Math #Statistics

$$x \sim \text{Laplace}(\mu,b), \\ f(x | \mu,b) =\frac{1}{2b} \exp (-\frac{|x-\mu|}{b}), \\ F(x | \mu,b) = \frac{1}{2} \exp (\frac{x - \mu}{b}) \text{ if } x \le \mu, 1 - \frac{1}{2} \exp (-\frac{x - \mu}{b}) \text{ if } x \ge \mu$$

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Normal Gaussian Distribution

#Math #Statistics

$$X \sim \mathcal{N}(\mu,\sigma^2) \\ f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp{[-\frac{(x-\mu)^{2}}{2\sigma^{2}}]}$$

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Poisson Binomial Distribution

#Math #Statistics

$$Pr(K = k) = \sum_{A \in F_{k}} \prod_{i \in A} p_{i} \prod_{j \in A_{c}} (1-p_{j})$$

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Poisson Distribution

#Math #Statistics

$$X \sim \pi(\mu) \\f(x)=\frac{\mu^{x}}{x!}e^{-\mu}\\ \text{Poisson Mean} \mu \\ \text{Poisson Variance}\sigma^2=\mu$$

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Student t-Distribution

#Math #Statistics

$$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}, \\ F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}, \\ F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}$$

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Uniform Distribution

#Math #Statistics

$$X \sim U(a,b) \\ f(x)=\frac{1}{b-a} \text{for} a \le x \le b \text{else} 0 \\ F(x)=\frac{x-a}{b-a} \text{for} a \le x \le b, 1 \text{for} x > b, 0 \text{for} x < a$$

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