|
24.5 Statistics`DiscreteDistributions`
This package gives you access to the most commonly used discrete statistical distributions. You can compute their densities, means, variances and other related properties. The distributions themselves are represented in the symbolic form name[ ,  , ... ]. Functions such as Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument.
Some statistical distributions from the package DiscreteDistributions.
Other discrete distributions from the package DiscreteDistributions. (See Section 3.2.)
Most of the common discrete statistical distributions can be understood by considering a sequence of "trials", each with two possible outcomes, say "success" and "failure".
The Bernoulli distribution BernoulliDistribution[p] is the probability distribution for a single trial in which success, corresponding to value 1, occurs with probability  , and failure, corresponding to value 0, occurs with probability  .
The binomial distribution BinomialDistribution[n, p] is the distribution of the number of successes that occur in  independent trials when the probability for success in an individual trial is  .
The geometric distribution GeometricDistribution[p] is the distribution of the total number of trials before the first success occurs in a sequence of trials where the probability for success in each individual trial is  .
The discrete uniform distribution DiscreteUniformDistribution[n] is the distribution of an experiment with  outcomes that occur with equal probabilities.
The Poisson distribution PoissonDistribution[mu] is the distribution of the number of points that lie in a unit interval, when the density of points is  .
Functions of statistical distributions.
The cumulative distribution function (cdf) at  for a discrete distribution is given by the sum of the probability density function (pdf) up to  . In this package the distributions are represented in symbolic form. PDF[dist, x] evaluates the density at  if  is a numerical value, and otherwise leaves the function in symbolic form. Similarly, CDF[dist, x] gives the cumulative distribution. Domain[dist] gives the domain of PDF[dist, x] and CDF[dist, x].
The quantile Quantile[dist, q] is effectively the inverse of the cdf. It gives the value of x at which CDF[dist, x] reaches q. The median is given by Quantile[dist, 1/2]; quartiles, deciles and percentiles can also be expressed as quantiles. Quantiles are used in constructing confidence intervals for statistical parameters.
The mean Mean[dist] is the expectation of the random variable distributed according to  and is usually denoted by  . The mean is given by  , where  is the pdf of the distribution. The variance Variance[dist] is given by  . The square root of the variance is called the standard deviation, and is usually denoted by  .
The Skewness[dist] and Kurtosis[dist] functions give shape statistics summarizing the asymmetry and the peakedness of a distribution, respectively. Skewness is given by  and kurtosis is given by  .
The function CharacteristicFunction[dist, t] is given by  . Each distribution has a unique characteristic function, which is sometimes used instead of the pdf to define a distribution.
Random[dist] gives pseudorandom numbers from the specified distribution. It can be used with a seed like other built-in forms of Random, as described in Section 24.1.
This loads the package.
In[1]:= <<Statistics`DiscreteDistributions`
Here is a symbolic representation of the binomial distribution for  trials, each having probability  of success.
In[2]:= bdist = BinomialDistribution[34, 0.3]
Out[2]= 
This is the mean of the distribution.
In[3]:= Mean[bdist]
Out[3]= 
You can get the equation for the mean by using symbolic variables as arguments.
In[4]:= Mean[BinomialDistribution[n, p]]
Out[4]= 
Here is the   quantile, which is equal to the median.
In[5]:= Quantile[bdist, 0.5]
Out[5]= 
This gives the characteristic function of the geometric distribution.
In[6]:= CharacteristicFunction[GeometricDistribution[p], x]
Out[6]= 
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT. | 
