Sample - Maple Help

Student[Statistics]

 Sample
 generate random sample

 Calling Sequence Sample(X, n, numeric_option, output_option)

Parameters

 X - algebraic; random variable n - positive integer; sample size numeric_option - (optional) equation of the form numeric=value where value is true or false output_option - (optional) equation of the form output=x where x is value, plot, or both

Description

 • The Sample command generates a random sample drawn from the distribution given by X.
 • The first parameter, X, can be a random variable, or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).
 • The second parameter, n, is the sample size. The function will return a newly created Vector of length n, filled with the sample values.
 • If the option output is not included or is specified to be output=value, then the function will return the generated sample as a Vector. If output=plot is specified, then the function will return a density plot of the input random variable together with a histogram of the sample. If output=both is specified, then both the value and the plot will be returned.

Computation

 • If X is a continuous random variable, or an expression that contains a floating point value, or an expression that contains a continuous random variable, then the sample is returned as floating point values. Otherwise, the sample is returned as exact values.
 • By default, the data are generated according to the rule above. To always generate data numerically, specify the numeric or numeric=true option.

Examples

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

Straightforward sampling of a random variable.

 > $X≔\mathrm{NormalRandomVariable}\left(0,1\right)$
 ${X}{≔}{\mathrm{_R}}$ (1)
 > $A≔\mathrm{Sample}\left(X,10\right)$
 ${A}{≔}\left[\right]$ (2)

You can check how well the generated data fit the input model by specifying the output=plot option and comparing the their graphs.

 > $\mathrm{Sample}\left(X,{10}^{5},\mathrm{output}=\mathrm{plot}\right)$

You can also sample an expression involving two random variables.

 > $Y≔\mathrm{NormalRandomVariable}\left(0,1\right)$
 ${Y}{≔}{\mathrm{_R0}}$ (3)
 > $\mathrm{Sample}\left(\mathrm{exp}\left(X\right)Y,10\right)$
 $\left[\right]$ (4)

Consider a discrete random variable.

 > $B≔\mathrm{PoissonRandomVariable}\left(3\right)$
 ${B}{≔}{\mathrm{_R1}}$ (5)
 > $\mathrm{Sample}\left(\frac{B}{\mathrm{\pi }},10\right)$
 $\left[\right]$ (6)

To always generate floating point value data, specify the numeric or numeric=true option.

 > $\mathrm{Sample}\left(\frac{B}{\mathrm{\pi }},10,\mathrm{numeric}\right)$
 $\left[\right]$ (7)

Use the output=both option to obtain both the value and plot of the generated data.

 > $\mathrm{dataset},\mathrm{graph}≔\mathrm{Sample}\left(B,100,\mathrm{output}=\mathrm{both}\right)$
 ${\mathrm{dataset}}{,}{\mathrm{graph}}{≔}\left[{?}\right]{,}{\mathrm{PLOT}}{}\left({\mathrm{...}}\right)$ (8)
 > $\mathrm{dataset}$
 $\left[\right]$ (9)
 > $\mathrm{graph}$

You can also compute the statistics of the generated data.

 > $C≔\mathrm{Sample}\left({X}^{2},{10}^{4}\right)$
 > $\mathrm{Mean}\left(C\right)$
 ${0.994862958212511}$ (10)
 > $\mathrm{Median}\left(C\right)$
 ${0.455147477689335}$ (11)
 > $\mathrm{Skewness}\left(C\right)$
 ${2.81421618454522}$ (12)
 > $\mathrm{Kurtosis}\left(C\right)$
 ${14.8537269935385}$ (13)
 > $\mathrm{Variance}\left(C\right)$
 ${1.96550597257815}$ (14)
 > $\mathrm{StandardDeviation}\left(C\right)$
 ${1.40196503971324}$ (15)
 > $\mathrm{Quantile}\left(C,0.6\right)$
 ${0.713125222514264}$ (16)

References

 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.
 Walker, Alastair J. New Fast Method for Generating Discrete Random Numbers with Arbitrary Frequency Distributions, Electronic Letters, 10, 127-128.
 Walker, Alastair J. An Efficient Method for Generating Discrete Random Variables with General Distributions, ACM Trans. Math. Software, 3, 253-256.

Compatibility

 • The Student[Statistics][Sample] command was introduced in Maple 18.