Equation Database

EQUATION LIST

math

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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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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