The empirical distribution is a discrete probability distribution with probability function generated from the given sample.

If opts is not given, then Maple assigns a probability to each variate present within the sample equal to the fraction of occurrences of that value within the entire sample. If opts is specified, then plist should be a list or rtable of probabilities (nonnegative real numbers adding up to 1), having equally many entries as sample; the ith entry of plist specifies the probability for the ith entry of sample.

If all the values in sample are positive integers, you can use the ProbabilityTable distribution instead.

Note that the EmpiricalDistribution command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$P≔\mathrm{Array}\left(\left[1\,1\,1\,2\,4\,4\,5.5\right]\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{EmpiricalDistribution}\left(P\right)\right)\:$

$\mathrm{ProbabilityFunction}\left(X\,1\right)$

$\frac{3}{7}$

$\mathrm{ProbabilityFunction}\left(X\,5\right)$

$0$

$\mathrm{ProbabilityFunction}\left(X\,5.5\right)$

$\frac{1}{7}$

$\mathrm{CDF}\left(X\,4\right)$

$\frac{6}{7}$

$\mathrm{Mean}\left(X\right)$

$2.64285714285714$

$\mathrm{Variance}\left(X\right)$

$2.908163265$

$Y≔\mathrm{RandomVariable}\left(\mathrm{EmpiricalDistribution}\left(⟨1\,2.5\,4\,5⟩\,'\mathrm{probabilities}'=\left[\frac{1}{3}\,\frac{1}{4}\,\frac{13}{60}\,\frac{1}{5}\right]\right)\right)$

$Y≔\mathrm{\_R0}$

$\mathrm{ProbabilityFunction}\left(Y\,\frac{5}{2}\right)$

$\frac{1}{4}$

$\mathrm{CDF}\left(Y\,4.5\right)$

$0.8000000000$

$\mathrm{Median}\left(Y\right)$

$\frac{5}{2}$

$\mathrm{Tally}\left(\mathrm{Sample}\left(Y\,{10}^6\right)\right)$

$\left[5.=199920\,1.=332622\,4.=216982\,2.50000000000000=250476\right]$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman, L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

The probabilities option was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.