How to |
Work with Statistical Distributions
Statistical distributions have applications in many fields, including the biological, social, and physical sciences. Mathematica
represents statistical distributions as symbolic objects. You can obtain properties, results, and random numbers for hundreds of built-in or custom distributions by applying built-in functions to the objects.
Statistical distributions are simply Mathematica
You can use the PDF
function to get the probability density function for the distribution:
You can get numeric results by inserting numbers for
Compute the density for numeric values of
Symbolic results can be used in other functions as well.
Here the density function is plotted for specified values of
You can directly obtain common properties such as the mean, variance, cumulative distribution function (CDF), and characteristic function using built-in functions.
This is the mean for a binomial distribution for 100 trials with success probability .3:
Like a PDF, a characteristic function uniquely defines a distribution.
Obtain the general formula for the characteristic function of a Cauchy distribution:
You can also compute more general expected values, which give the value expected for a given function applied to a random variable from a given distribution. The
raw moment is the expected value of
raised to the
raw moment for a Poisson-distributed random variable
You can generate random numbers from distributions using RandomVariate
These are 10 numbers simulated from a
distribution with 15 degrees of freedom:
A geometric distribution describes the number of trials before a failure when there is a probability
of success in each trial.
Simulate 20 numbers from a geometric distribution with success probability parameter
You could even visualize a sample against a theoretical distribution because plots of data and functions can be combined.
gamma-distributed numbers are generated and stored to the symbol data
You can use Histogram
to generate a histogram of these values on a probability density scale:
You can visualize the theoretical density function using Plot
You can then use Show
to display the two graphics together:
You might also want to estimate parameter values assuming a dataset follows a particular distribution. For instance, you could find the maximum likelihood estimate for parameters by using FindDistributionParameters
The results can be packaged up into a distribution object using EstimatedDistribution
The log-likelihood could also be computed using LogLikelihood
with the estimated distribution:
The log-likelihood value is mostly relevant compared to log-likelihood values for other parameters. Creating a ContourPlot
near the obtained values can provide a qualitative comparison. Points on a given contour have the same log-likelihood.
Here a white point is placed at the optimal point: