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Matlab Distributions

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Probability Distributions using Matlab

Probability Distributions using Matlab

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Matlab Distributions Matlab 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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