The empirical random variable 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.

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

$P≔\left[1\,1\,1\,2\,2\,2.5\,3\,4\,4\,5.5\right]\:$

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

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

$\frac{3}{10}$

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

$0$

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

$\frac{1}{10}$

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

$\mathrm{CDF}\left(X\,\frac{9}{2}\right)$

$\frac{9}{10}$

$\mathrm{CDF}\left(X\,x\,\mathrm{output}=\mathrm{plot}\right)$

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

$2.60000000000000$

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

$2.090000000$

$M≔\mathrm{Matrix}\left(\left[\left[1\,2\,3\right]\,\left[3.5\,4\,5\right]\right]\right)$

$M≔\left[\begin{array}{ccc}1& 2& 3\\ 3.5& 4& 5\end{array}\right]$

$\mathrm{pmatrix}≔\mathrm{Matrix}\left(\left[\left[\frac{1}{15}\,\frac{2}{15}\,\frac{5}{15}\right]\,\left[\frac{4}{15}\,\frac{1}{15}\,\frac{2}{15}\right]\right]\right)$

$\mathrm{pmatrix}≔\left[\begin{array}{ccc}\frac{1}{15}& \frac{2}{15}& \frac{1}{3}\\ \frac{4}{15}& \frac{1}{15}& \frac{2}{15}\end{array}\right]$

$Y≔\mathrm{EmpiricalRandomVariable}\left(M\,\mathrm{probabilities}=\mathrm{pmatrix}\right)\:$

$\mathrm{ProbabilityFunction}\left(Y\,x\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{ProbabilityFunction}\left(Y\,3.5\right)$

$\frac{4}{15}$

$\mathrm{CDF}\left(Y\,x\,\mathrm{output}=\mathrm{plot}\right)$

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

$\frac{13}{15}$

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

$3$

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 Student[Statistics][EmpiricalRandomVariable] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.