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Matlab: Statistics and Distributions
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Matlab: Statistics and Distributions

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Statistics and Distributions using Matlab

Statistics and Distributions using Matlab

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    Matlab: Statistics and Distributions Matlab: Statistics and Distributions Presentation Transcript

    • Distributions
    • Probability Distributions
      Wikipedia: A probability distribution identifies either the probability of each value of an unidentified random variable (when the variable is discrete), or the probability of the value falling within a particular interval (when the variable is continuous). The probability distribution describes the range of possible values that a random variable can attain and the probability that the value of the random variable is within any (measurable) subset of that range.
    • Types of supported distributions
      pdf — Probability density functions
      cdf — Cumulative distribution functions
      inv — Inverse cumulative distribution functions
      stat — Distribution statistics functions
      fit — Distribution fitting functions
      like — Negative log-likelihood functions
      rnd — Random number generators
    • Supported Distributions
      Bernoulli Distribution
      Beta Distribution
      Binomial Distribution
      Birnbaum-Saunders Distribution
      Chi-Square Distribution
      Copulas
      • Geometric Distribution
      • Hypergeometric Distribution
      • Inverse Gaussian Distribution
      • Inverse Wishart Distribution
      • Johnson System
      • Logistic Distribution
      • Loglogistic Distribution
      • Custom Distributions
      • Exponential Distribution
      • Extreme Value
      • Distribution
      • F Distribution
      • Gamma Distribution
      • Gaussian Distribution
      • Gaussian Mixture Distributions
      • Generalized Extreme Value Distribution
      • Generalized Pareto Distribution
    • Supported Distributions
      Lognormal Distribution
      Multinomial Distribution
      Multivariate Gaussian Distribution
      Multivariate Normal Distribution
      Multivariate t Distribution
      Nakagami Distribution
      Negative Binomial Distribution
      Noncentral Chi-Square Distribution
      Noncentral F Distribution
      Noncentral t Distribution
      Nonparametric Distributions
      • Normal Distribution
      • Pareto Distribution
      • Pearson System
      • Piecewise Distributions
      • Poisson Distribution
      • Rayleigh Distribution
      • Rician Distribution
      • Student's t Distribution
      • t Location-Scale Distribution
      • Uniform Distribution (Continuous)
      • Uniform Distribution (Discrete)
      • Weibull Distribution
      • Wishart Distribution
    • Probability Density functions
      Parametric Estimation
      Nonparametric Estimation
    • 1. Parametric estimation
      • p = 0.2; % Probability of success for each trial
      • n = 10; % Number of trials
      • k = 0:n; % Outcomes
      • m = binopdf(k,n,p); % Probability mass vector
      • bar(k,m) % Visualize the probability distribution
      • set(get(gca,'Children'),'FaceColor',[.8 .8 1])
      • grid on
    • 1. Parametric estimation
    • 2. Non parametric estimation
      A distribution of data can be described graphically with a histogram:
      • cars = load('carsmall','MPG','Origin');
      • MPG = cars.MPG;
      • hist(MPG)
      • set(get(gca,'Children'),'FaceColor',[.8 .8 1])
    • 2. Non parametric estimation
    • 2. Non parametric estimation
      You can also describe a data distribution by estimating its density. The ksdensity function does this using a kernel smoothing method. A nonparametric density estimate of the data above, using the default kernel and bandwidth, is given by:
      • [f,x] = ksdensity(MPG);
      • plot(x,f);
      • title('Density estimate for MPG') ;
    • 2. Non parametric estimation
    • Cumulative Distribution Functions
      Parametric Estimation
      Nonparametric Estimation
    • Inverse Cumulative Distribution Functions
      Each function in this family represents a parametric family of distributions. Input arguments are arrays of cumulative probabilities between 0 and 1 followed by a list of parameter values specifying a particular member of the distribution family.
    • Inverse Cumulative Distribution Functions
      The expinv function can be used to compute inverses of exponential cumulative probabilities:
    • Distribution Statistics Functions
      Each function in this family represents a parametric family of distributions. Input arguments are lists of parameter values specifying a particular member of the distribution family. Functions return the mean and variance of the distribution, as a function of the parameters.
    • Distribution Statistics Functions
      For example, the wblstat function can be used to visualize the mean of the Weibull distribution as a function of its two distribution parameters:
      • a = 0.5:0.1:3;
      • b = 0.5:0.1:3;
      • [A,B] = meshgrid(a,b);
      • M = wblstat(A,B);
      • surfc(A,B,M)
    • Distribution Statistics Functions
    • Distribution Fitting Functions
      Fitting Supported Distributions
      Fitting Piecewise Distributions
    • Negative Log-Likelihood Functions
      Each function in this family represents a parametric family of distributions. Input arguments are lists of parameter values specifying a particular member of the distribution family followed by an array of data. Functions return the negative log-likelihood of the parameters, given the data.
    • Random Number Generators
      Each RNG represents a parametric family of distributions. Input arguments are lists of parameter values specifying a particular member of the distribution family followed by the dimensions of an array. RNGs return random numbers from the specified distribution in an array of the specified dimensions.
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