List of Statistics Equations Latex Code

rockingdingo 2024-01-09 #Binomial #Poisson #Normal Gaussian #Covariance #Chi-Square

List of Statistics Equations Latex Code

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In this blog, we will summarize the latex code for statistics equations, ranging from the elementary statistics equations (e.g. mean and variance) to more advanced graduate-level statistics equations, e.g. Binomial, Poisson, Normal Distribution, Chi-Square Test, Skewness, Kurtosis, etc.

• 1. Elementary Statistics
• 1.1 Mean and Variance
• 1.2 Standard Deviation
• 2. Probability Distributions
• 2.1 Binomial Distribution
• 2.2 Poisson Distribution
• 2.3 Normal Gaussian Distribution
• 2.4 Chi-Square Test
• 3. Advanced Statistics
• 3.1 Sum of Random Variables
• 3.2 Covariance
• Gamma Distribution
• Fisher Skewness
• Kurtosis

1. Elementary Statistics

• Mean and Variance

  Equation

  
  $$\text{Mean Discrete}\ \mu=E(X)=\sum P_{i}x_{i}$$
  $$\text{Mean Continuous}\ \mu=\int xf(x) dx$$
  $$\text{Variance Discrete}\ \sigma^{2}=V(X)=E[(X-\mu)^2]=\sum P_{i}(x_{i} -\mu)^2$$
  $$\text{Variance Continuous}\ \sigma^{2}=V(X)=E[(X-\mu)^2]=\int (x-\mu)^{2}f(x) dx$$
  

  Latex Code

          \text{Mean Discrete}\ \mu=E(X)=\sum P_{i}x_{i} \\\text{Mean Continuous}\ \mu=\int xf(x) dx \\\text{Variance Discrete}\ \sigma^{2}=V(X)=E[(X-\mu)^2]=\sum P_{i}(x_{i} -\mu)^2 \\\text{Variance Continuous}\ \sigma^{2}=V(X)=E[(X-\mu)^2]=\int (x-\mu)^{2}f(x) dx
          

  Explanation

  X denotes a random variable which has a distribution f(x) over some subset x of the real numbers. If the distribution f(x) is discrete, the probability of f(x=X)=xi is is Pi. And the mean \mu equals to the sum of probability Pi multiplies the random variable value x. . When the distribution is continuous f(x), the probability that X lies in the interval, and the f(x) denotes density function. The variance(squared value of standard deviation \sigma) measure how far the subset X is from the mean value \mu, is it a flat distribution or shallow distribution. And the definition of variance is the expectation E(X) of the squared distance between each data point X and its mean \mu.

• Standard Deviation

  Equation

  
  $$\sigma=\sqrt{V(X)}=\sqrt{\sum_{i} P_{i}(x_{i} - \mu)^2}=\sqrt{\frac{\sum_{i} (x_{i} - \mu)^2}{n}}$$
  $$\sigma=\sqrt{V(X)}=\sqrt{\int (x-\mu)^{2}f(x) dx}$$
  

  Latex Code

          \sigma=\sqrt{V(X)}=\sqrt{\sum_{i} P_{i}(x_{i} - \mu)^2}=\sqrt{\frac{\sum_{i} (x_{i} - \mu)^2}{n}} \\ \sigma=\sqrt{V(X)}=\sqrt{\int (x-\mu)^{2}f(x) dx}
          

  Explanation

  Standard Deviation \sigma equals to the squared root of Variance \sigma^{2} of a dataset X.

2. 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

  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

  \mu equals to the probability that an event occurs in a unit time period.

• Normal Gaussian Distribution

  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

  X denotes the random variable which follows the normal distribution. \mu denotes the mean value and \sigma denotes the standard deviation.

• Chi-Square Test

  Equation

  $$\chi ^{2}=\sum \frac{(A-E)^2}{E}$$
  $$=\sum^{K}_{1}\frac{(A_{i}-E_{i})^2}{E_{i}}=\sum^{K}_{1}\frac{(A_{i}-np_{i})^2}{np_{i}}$$
  

  Latex Code

          \chi ^{2}=\sum \frac{(A-E)^2}{E}\\=\sum^{K}_{1}\frac{(A_{i}-E_{i})^2}{E_{i}}=\sum^{K}_{1}\frac{(A_{i}-np_{i})^2}{np_{i}}
          

  Explanation

  Chi-Square Test measure how close and an actual observation of distribution A are correlated to the assumed theoretical distribution E. The dataset X are splitted into K different buckets and the statistics of Chi-Square Test is calculated as above.

3. Advanced Statistics

• Sum of Random Variables

  Equation

  $$W=aX+bY$$
  $$E(W)=aE(X)+bE(Y)$$
  $$\text{Var}(W)=a^{2}\text{Var}(X) + b^{2}\text{Var}(Y), \text{X and Y independent}$$
  

  Latex Code

          W=aX+bY \\ E(W)=aE(X)+bE(Y) \\ \text{Var}(W)=a^{2}\text{Var}(X) + b^{2}\text{Var}(Y), \text{X and Y independent}
          

  Explanation

  E(X) and E(Y) denote the mean of random variable X and Y, Var(X) and Var(Y) denote the variance of random variable X and Y. W is the weighted sum of two random variables.

• Covariance

  Equation

  $$\text{Cov}(X,Y)=E[(X-E(X))(Y-E(Y))]$$
  $$=E(XY)-2E(X)E(Y)+E(X)E(Y)$$
  $$=E(XY)-E(X)E(Y)$$
  

  Latex Code

          \text{Cov}(X,Y)=E[(X-E(X))(Y-E(Y))]\\=E(XY)-2E(X)E(Y)+E(X)E(Y)\\=E(XY)-E(X)E(Y)
          

  Explanation

  Covariance measures the total variation of two random variables X and Y from their expected values E(X) and E(Y). The definition of covariance is E[(X-E(X))(Y-E(Y))].

• Gamma Distribution

  Math,Statistics

  Equation

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

  Latex Code

              \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}
          

  Explanation

  Latex code for Gamma Distribution. A gamma distribution is a general type of statistical distribution that is related to the beta distribution and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. Gamma distributions have two free parameters, labeled alpha and theta. We now let W denote the waiting time until the a-th event occurs and find the distribution of W. We could represent the situation as follows:

  • $$\Gamma(a)$$ : Gamma function of a
  • $$\alpha$$: Gamma function parameter
  • $$P(x) = \frac{x^{\alpha-1} e^{-frac{x}{\theta}}}{\Gamma(\alpha) \theta^{\alpha}}$$: PDF of gamma distributed random variable X.
  • $$\mu = \alpha \theta$$: Mean of Gamma Distribution
  • $$\sigma^{2} = \alpha \theta^{2}$$: Variance of Gamma Distribution
  • $$\gamma_{1} = \frac{2}{\sqrt{\alpha}}$$: Skewness of Gamma Distribution
  • $$\gamma_{2} = \frac{6}{\alpha}$$: Kurtosis of Gamma Distribution
  
  

  Related Documents

  • Gamma Distribution
  • 15.4 - Gamma Distributions

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• Fisher Skewness

  Math,Statistics

  Equation

  Skewness measures $$ g_{1} = \frac{\sum_{i=1}^{n}{(x_{i} - \bar{x})^{3}}/n} {s^{3}}$$ The adjusted Fisher-Pearson skewness coefficient is: $$G_{1} = \frac{\sqrt{n(n-1)}}{n-2} \frac{\sum_{i=1}^{n}{(x_{i} - \bar{x})}/n} {s^{3}}$$
  

  Latex Code

              g_{1} = \frac{\sum_{i=1}^{n}{(x_{i} - \bar{x})^{3}}/n} {s^{3}} \\
              G_{1} = \frac{\sqrt{n(n-1)}}{n-2} \frac{\sum_{i=1}^{n}{(x_{i} - \bar{x})}/n} {s^{3}}
          

  Explanation

  Latex code for Fisher Skewness. Skewness measures the lack of symmetry in a variable. The formula for the Fisher-Pearson skewness coefficient is: The formula for the Fisher-Pearson skewness coefficient is:

  • Fisher-Pearson skewness coefficient: $$g_{1} = \frac{\sum_{i=1}^{n}{(x_{i} - \bar{x})^{3}}/n} {s^{3}}$$
  • Adjusted Fisher-Pearson skewness coefficient: $$G_{1} = \frac{\sqrt{n(n-1)}}{n-2} \frac{\sum_{i=1}^{n}{(x_{i} - \bar{x})}/n} {s^{3}}$$
  
  

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  • SKEWNESS

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• Kurtosis

  Math,Statistics

  Equation

  $$\mbox{kurtosis} = \frac{\sum_{i=1}^{N}(Y_{i} - \bar{Y})^{4}/N} {s^{4}}$$

  Latex Code

              \mbox{kurtosis} = \frac{\sum_{i=1}^{N}(Y_{i} - \bar{Y})^{4}/N} {s^{4}}
          

  Explanation

  For univariate data Y1, Y2, ..., YN, the formula for kurtosis is as below, Note that in computing the kurtosis, the standard deviation is computed using N in the denominator rather than N - 1.

  • $$\bar{Y}$$: Mean
  • $$s$$: Standard deviation
  • $$N$$: Number of data points
  
  

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  • NIST HANDBOOK Kurtosis

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