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In this blog, we will summarize the latex code for Probability Formulas and Equations, including Binomial Distribution, Poisson Distribution, Normal Gaussian Distribution, Exponential Distribution, Gamma Distribution, Uniform Distribution, Beta Distribution, Bernoulli Distribution, Geometric Distribution, Beta Binomial Distribution, Poisson Binomial Distribution, Chi-Squared Distribution, Gumbel Distribution, Student t-Distribution, Laplace Distribution, etc. And for multivariate distributions, we will also cover Multinomial Distribution, MultiVariate Normal Distribution, MultiVariate Gamma Distribution, MultiVariate t-Distribution and others.

• 1. Probability Distributions
• Binomial Distribution
• Poisson Distribution
• Normal Gaussian Distribution
• Exponential Distribution
• Gamma Distribution
• Uniform Distribution
• Beta Distribution
• Bernoulli Distribution
• Geometric Distribution
• Beta Binomial Distribution
• Poisson Binomial Distribution
• Chi-Squared Distribution
• Gumbel Distribution
• Student t-Distribution
• Laplace Distribution
• 2. Multivariate Distributions
• Multinomial Distribution
• MultiVariate Normal Distribution
• MultiVariate Gamma Distribution
• MultiVariate t-Distribution

1. Probability Distributions

• Binomial Distribution

  Equation

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

  Latex Code

              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
          

  Explanation

  The binomial distribution measures in total n independent trials, the probability that x trials in total n trials are positive (like the getting positive of flipping a coin). In this formulation, f(x) denotes the probability that x positive trials are observed in n independent trials. p denote the probability that positive is observed in each single trial. q denotes the negative is observed, which equals to 1-p.

• Poisson Distribution

  Math,Statistics

  Equation

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

  Latex Code

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

  Explanation

  Latex code for Poisson Distribution.

  • The probability rate that an event occurs in a unit time period $$\mu$$
  • PDF of Poisson Distribution: $$f(x)=\frac{\mu^{x}}{x!}e^{-\mu}$$
  • CDF of Poisson Distribution: $$F(x) = \frac{\Gamma([k+1], \lambda)}{[k]!} $$
  • Mean of Poisson Distribution: $$\mu$$
  • Variance of Poisson Distribution: $$\mu$$

• Normal Gaussian Distribution

  Math,Statistics

  Equation

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

  Latex Code

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

  Explanation

  Latex code for Normal Gaussian Distribution.

• PDF of Normal Gaussian Distribution: $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp{[-\frac{(x-\mu)^{2}}{2\sigma^{2}}]}$$
• Mean of Normal Gaussian Distribution: $$\mu$$
• Variance of Normal Gaussian Distribution: $$\sigma^{2}$$

• Exponential Distribution

  Math,Statistics

  Equation

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

  Latex Code

              X \sim \exp(\lambda), \\
              f(x, \lambda)=\lambda e^{-\lambda x}, \\
              F(x, \lambda)=1 - e^{-\lambda x}
          

  Explanation

  Latex code for Exponential Distribution.

  • The rate parameter: $$\lambda=\frac{1}{\mu}$$
  • Constant Exponential Parameter: $$e$$
  • PDF of Exponential Distribution: $$f(x, \lambda)=\lambda e^{-\lambda x}$$
  • CDF of Exponential Distribution: $$F(x, \lambda)=1 - e^{-\lambda x}$$
  • Mean of Exponential Distribution: $$\frac{1}{\lambda}$$
  • Variance of Exponential Distribution: $$\frac{1}{\lambda^{2}}$$

• Gamma Distribution

  Math,Statistics

  Equation

  
  $$X \sim \Gamma(\alpha,\beta) = \text{Gamma}(\alpha,\beta)$$ $$k = \alpha, \beta=\frac{1}{\theta} $$ $$f(x)=\frac{1}{\Gamma (k)\theta^{k}}x^{k-1}e^{-\frac{x}{\theta}}, f(x)=\frac{\beta^{\alpha}}{\Gamma (\alpha)}x^{\alpha-1}e^{-\beta x}$$ $$F(x)=\frac{1}{\Gamma (k)}\gamma(k,\frac{x}{\theta}), F(x)=\frac{1}{\Gamma (\alpha)}\gamma(\alpha,\beta x)$$
  

  Latex Code

              X \sim \Gamma(\alpha,\beta) = \text{Gamma}(\alpha,\beta) \\
              k = \alpha, \beta=\frac{1}{\theta} \\
              f(x)=\frac{1}{\Gamma (k)\theta^{k}}x^{k-1}e^{-\frac{x}{\theta}}, f(x)=\frac{\beta^{\alpha}}{\Gamma (\alpha)}x^{\alpha-1}e^{-\beta x} \\
              F(x)=\frac{1}{\Gamma (k)}\gamma(k,\frac{x}{\theta}), F(x)=\frac{1}{\Gamma (\alpha)}\gamma(\alpha,\beta x)
          

  Explanation

  Latex code for the Gamma Distribution.

  • Shape parameter: $$k$$
  • Scale parameter: $$\theta$$
  • Shape parameter: $$\alpha=k$$:
  • Rate parameter: $$\beta=\frac{1}{\theta}$$
  • PDF for Gamma Distribution: $$f(x)=\frac{1}{\Gamma (k)\theta^{k}}x^{k-1}e^{-\frac{x}{\theta}}$$
  • CDF for Gamma Distribution: $$F(x)=\frac{1}{\Gamma (k)}\gamma(k,\frac{x}{\theta})$$
  • Mean for Gamma Distribution: $$k\theta,\frac{\alpha}{\beta}$$
  • Variance for Gamma Distribution: $$k\theta^2,\frac{\alpha}{\beta^2}$$
  
  

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  • Wikipedia Probability Distribution

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• Uniform Distribution

  Math,Statistics

  Equation

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

  Latex Code

              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 
          

  Explanation

  Latex code for the Uniform Distribution.

  • Lower Bound parameter: $$a$$
  • Upper Bound parameter: $$b$$
  • PDF for Uniform Distribution: $$f(x)=\frac{1}{b-a} \text{for} a \le x \le b \text{else} 0$$
  • CDF for Uniform Distribution: $$F(x)=\frac{x-a}{b-a} \text{for} a \le x \le b, 1 \text{for} x > b, 0 \text{for} x < a$$
  • Mean for Uniform Distribution: $$\frac{a+b}{2}$$
  • Variance for Uniform Distribution: $$\frac{(b-a)^{2}}{12}$$
  
  

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• Beta Distribution

  Math,Statistics

  Equation

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

  Latex Code

              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)
          

  Explanation

  Latex code for the Beta Distribution.

  • Shape parameter: $$\alpha, \beta$$
  • PDF for Beta Distribution: $$f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$$
  • Beta Function: $$B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$$
  • CDF for Beta Distribution, which is the regularized incomplete beta function: $$F(x)=I_{x}(\alpha+\beta)$$
  • Mean for Beta Distribution: $$E[X] = \frac{\alpha}{\alpha+\beta}$$
  • Variance for Beta Distribution: $$var[X] = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} $$
  
  

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• Bernoulli Distribution

  Math,Statistics

  Equation

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

  Latex Code

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

  Explanation

  Latex code for the Bernoulli Distribution.

  • Probability parameter: $$p, q$$
  • PDF for Bernoulli Distribution: $$f(x)=p \text{ if } k = 1 \text{ else } q \text{ if } k = 0$$
  • CDF for Bernoulli Distribution: $$0 \text{ if } k < 0, 1-p \text{ if } 0 \le k \lt 1, 1 \text{ if } k \ge 1$$
  
  

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• Geometric Distribution

  Math,Statistics

  Equation

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

  Latex Code

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

  Explanation

  Latex code for the Geometric Distribution.

  • Probability parameter p means the success probability of each trial: $$p$$
  • The number of total trials until the first successful trial k: $$k$$
  • PDF for Geometric Distribution: $$f(x)=(1-p)^{k-1}q$$
  • CDF for Geometric Distribution: $$F(x)=1 - (1-p)^{[x]}$$
  
  

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• Beta Binomial Distribution

  Math,Statistics

  Equation

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

  Latex Code

              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)}
          

  Explanation

  Latex code for the Beta Binomial Distribution. Beta Binomial Distribution describe the situation that the probability of success in each trial is heteregenous or different as $$\text{Beta}(p|\alpha,\beta)$$.

  • Beta Binomial Distribution: $$x \sim f(x | n, \alpha, \beta)$$
  • Hyperparameters of $$\alpha,\beta$$ of beta distribution
  • Binomial trials $$C^{x}_{n}$$
  
  

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• Poisson Binomial Distribution

  Math,Statistics

  Equation

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

  Latex Code

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

  Explanation

  Latex code for the Poisson Binomial Distribution. Poisson Binomial Distribution measures the probability of having k successful trials out of a total of n can be written as the sum. The success probability of each trial p1,p2,pn are not identical as the standard binomial distribution.

  • The set of all subsets of k integers that can be selected from set {1,2,...,n} denotes: $$F_{k}$$
  • The set of success trials : $$A$$
  • The set of failed trials, which is also complement of set A: $$A^{c}$$
  
  

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• Chi-Squared Distribution

  Math,Statistics

  Equation

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

  Latex Code

              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})
          

  Explanation

  Latex code for the Chi-Squared Distribution. Chi-Squared distribution describes the situation that random variable Q is the sum of k squared value of independent standard normal random variables Z_{1} to Z_{k}.

  • Chi-Squared distribution definition: $$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k)$$
  • PDF of Chi-Squared distribution: $$\frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2} $$
  • CDF of Chi-Squared distribution: $$\frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2}) $$
  • Mean value of Chi-Squared distribution: $$k$$
  • Variance value of Chi-Squared distribution: $$2k$$
  
  

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• Gumbel Distribution

  Math,Statistics

  Equation

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

  Latex Code

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

  Explanation

  Latex code for the Gumbel Distribution. The Gumbel Distribution is used to model the distribution of the maximum of a number of samples of various distributions.

  • PDF of Gumbel distribution: $$\frac{1}{\beta} e^{-(z + e^{-z})}, z=\frac{x - \mu}{\beta}$$
  • CDF of Gumbel distribution: $$e^{-e^{-(x-\mu)/\beta}}$$
  • Mean value of Gumbel distribution: $$\mu +\beta\gamma$$
  • Variance value of Gumbel distribution: $$\frac{\pi^{2}}{6}\beta^{2}$$
  
  

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• Student t-Distribution

  Math,Statistics

  Equation

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

  Latex Code

              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}
          

  Explanation

  Latex code for the Student t-Distribution. The Student t-Distribution t_{v} is a continuous distribution that generalize the standard normal distribution. But the t-Distribution t_{v} has heavier tails.

  • PDF of Student t-Distribution: $$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}$$
  • CDF of Student t-Distribution: $$F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}$$
  • Hypergeometric function : $$2^{F_{1}}(a,b;c;z)$$
  
  

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• Laplace Distribution

  Math,Statistics

  Equation

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

  Latex Code

              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
          

  Explanation

  Latex code for Laplace Distribution. Laplace Distribution is also called double exponential distribution. It can be thought of as two exponential distributions spliced together. \mu denotes the location parameters and b is the scale parameter.

  • PDF of Laplace Distribution: $$f(x | \mu,b) =\frac{1}{2b} \exp (-\frac{|x-\mu|}{b})$$
  • CDF of Laplace Distribution: $$F(x | \mu,b) = \frac{1}{2} \exp (\frac{x - \mu}{b}) \text{ if } x \le \mu, 1 - \frac{1}{2} \exp (-\frac{x - \mu}{b}) \text{ if } x \ge \mu$$
  
  

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