Estimate both parameters for a binomial distribution:

Estimate p, assuming n is known:

Estimate n, assuming p is known:

Get the distribution with maximum likelihood parameter estimate for a particular family:

Check goodness of fit by comparing a histogram of the data and the estimate's PDF:

Perform goodness-of-fit tests with null distribution from res:

Perform tests correcting for estimation of the parameter:

Estimate parameters by maximizing the log‐likelihood:

Plot the log‐likelihood function to visually check that the solution is optimal:

Visualize a log‐likelihood surface to find rough values for the parameters:

Supply those rough values as starting values for the estimation:

Mark the optimal point on the contour plot:

Estimate the normal approximation of Poisson data:

Obtain estimate to 20 digits:

Estimate parameters for a continuous distribution:

Estimate parameters for a discrete distribution:

Compare the fitted and empirical CDFs:

Estimate parameters for a discrete multivariate distribution:

Estimate parameters for a continuous multivariate distribution:

Visualize the density functions for the marginal distributions:

Obtain the covariance matrix from the formula:

Estimate parameters for a truncated normal:

Estimate parameters for a constructed distribution:

Visualize the optimal point:

Estimate parameters for a product distribution:

Estimate parameters for a copula distribution:

Estimate parameters for a component mixture:

Estimate the mixture probabilities assuming the component distributions are known:

Visualize the two estimates against the data:

Estimate parameters for a distribution in specified units:

Estimate parameters by matching cumulants:

Other moment‐based methods typically give similar results:

Estimate parameters based on default moments:

Estimate parameters from the first and fourth moments:

Obtain the maximum likelihood estimates using the default method:

Use FindMaximum to obtain the estimates:

Use EvaluationMonitor to extract the points sampled:

Visualize the sequences of sampled and values:

Use machine precision for continuous parameters by default:

Obtain a higher-precision result:

Get the method of moments estimate:

Use the method of moments estimate as the starting value for ml estimation:

Obtain ml estimates for a gamma distribution:

Use those as starting values for the method of moments:

Estimate Laplace parameters for data from an ExponentialPowerDistribution:

Use the Laplace estimate as a starting point for estimating exponential power parameters:

Compare the data with the Laplace and exponential power estimates:

Model lognormal distributed data with a gamma distribution:

Compare the distributions of the simulation and estimated distributions:

The number of accident claims per policy per year from an insurance company:

Estimate the parameter for a logarithmic series distribution for policy claims shifted by 1:

See that the estimate gives a maximal result:

Get word length data for several languages:

Model the word lengths for each language as binomially distributed with :

Compare the actual and estimated distributions:

Bootstrap the distribution of p values based on these 9 results:

Estimate the expected value of p and a standard deviation for the estimate:

The word count in a text follows a Zipf distribution:

Fit a ZipfDistribution to the word frequency data:

Fit a truncated ZipfDistribution to counts at most 50 using rhohat as a starting value:

Visualize the CDFs up to the truncation value:

Estimate the proportion of the original data not included in the truncated model:

Find estimates for a multimodal MixtureDistribution model:

The magnitudes of earthquakes in the United States in the selected years have two modes:

Fit distribution from possible mixtures of one NormalDistribution with another:

Extract the means of the components:

The components' means are far enough apart that they are still the modes:

Model monthly maximum wind speeds in Boston:

Fit the data to a RayleighDistribution:

An ExtremeValueDistribution:

Compare the empirical and fitted quantiles to see where the models deviate from the data:

Model incomes at a large state university:

Assume the salaries are Dagum distributed:

Assume they follow a more general Pareto distribution:

Compare the subtle differences in the estimated distributions:

Use a beta distribution to model the proportion of Dow Jones Industrial stocks that increase in value on a given day:

Find daily change for Dow Jones Industrial stocks:

Number of days for each financial entity:

Extract values from time series for each entity and normalize numeric quantities:

Check if each entity has the same length of data:

Calculate the daily ratio of companies with an increase in value:

Find parameter estimates, excluding days with zero or all companies having an increase in value:

Visualize the likelihood contours and mark the optimal point:

The average city and highway mileage for midsize cars follows a binormal distribution:

Assume city and highway miles per gallon are normally distributed and correlated:

Extract the estimated average city and highway mileages:

Extract the estimated correlation between city and highway mileages:

Visualize the joint density on a logarithmic scale with the mean mileage marked with a blue point:

The data contains waiting times in days between serious (magnitude at least 7.5 or over 1000 fatalities) earthquakes worldwide, recorded from 12/16/1902 to 3/4/1977:

Model waiting times by an ExponentialDistribution:

Estimate the average and median number of days between major earthquakes:

The number of earthquakes per year can be modeled by SinghMaddalaDistribution:

Fit the distribution to the data:

Compute the maximized log‐likelihood:

Visualize the log‐likelihood profiles near the optimal parameter values:

Mixtures can be used to model multimodal data:

A histogram of waiting times for eruptions of the Old Faithful geyser exhibits two modes:

Fit a mixture of gamma and normal distributions to the data:

Compare the histogram to the PDF of the estimated distribution:

Lognormal distribution can be used to model stock prices:

Fit the distribution to the data:

Visualize the profile likelihoods, fixing one parameter at the fitted value:

Consider the annual minimum daily flows given in cubic meters per second for the Mahanadi river:

Model the annual minimum mean daily flows as a MinStableDistribution:

Simulate annual minimum mean daily flows for the next 30 years:

Use a Pareto distribution to model Australian city population sizes:

Get the probability that a city has a population at least 10000 under a Pareto distribution:

Compute the probability given the parameter estimates:

Compute the probability based on the original data: