List of Statistics Equations Latex Code

rockingdingo 2022-11-06 #Binomial #Poisson #Normal Gaussian #Covariance #Chi-Square
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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, etc.

    1. Elementary Statistics

  • Mean and Variance

    Equation


    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


    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


    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


    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


    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


    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


    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


    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))].




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