stats::empiricalPF

Probability function of a finite data sample

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

stats::empiricalPF(x1, x2, …)
stats::empiricalPF([x1, x2, …])
stats::empiricalPF(n, <c>)
stats::empiricalPF(n, <[c]>)

Description

stats::empiricalPF([x1, x2, …, xn]) returns a procedure representing the probability function

of the sample given by the data x1, x2, ….

The procedure f := stats::empiricalPF([x1, x2, …]) can be called in the form f(x) with an arithmetical expression x or sets of lists of such expressions.

If x is a numerical expression that is contained in the data x1, x2, …, then the corresponding probability value is returned (n is the size of the sample).

If x is a numerical expression that is not contained in the data x1, x2, …, then 0 is returned.

If x is a symbolic expression that cannot be converted to a real floating-point number, f(x) returns the symbolic call stats::empiricalPF([x1, x2, …])(x) with the data x1, x2, … in ascending order.

If x is a set, the sum of the probability values of its elements is returned.

If x is a list, it is treated like a set (i.e., duplicate entries in x are eliminated). The sum of the probability values of the elements in x is returned.

Duplicate data elements are automatically combined to a single data element, adding up the corresponding probability values. Cf. Example 4.

stats::empiricalPF is generalized by stats::finitePF, which allows to specify different probabilities for the elements of the sample. The call stats::empiricalPF([x_1, dots, x_n], [1/n, dots, 1/n]) corresponds to stats::empiricalPF([x1, …, xn]).

Further, stats::finitePF does not only allow numerical values x1, x2, …, but arbitrary MuPAD® objects.

Examples

Example 1

We demonstrate the basic usage of this function:

f := stats::empiricalPF(1, 3, PI, 4.0):
f(0), f(1), f(1.0), f(3), f(PI), f(float(PI)), f(4), f(4.0)

Alternatively, the data may be passed as a list:

f := stats::empiricalPF(1, 3, PI, 4.0):
f(0), f(1), f(1.0), f(3), f(PI), f(float(PI)), f(4), f(4.0)

A symbolic value of the argument in f leads to a symbolic return value:

f(x)

Symbolic data are not accepted:

stats::empiricalPF(1, 3, x, 4.0):
Error: Some data cannot be converted to floating-point numbers. [stats::empiricalPF]
delete f:

Example 2

We create a sample of type stats::sample consisting of one string column and two non-string columns:

s := stats::sample(
  [["1996", 1242, 2/5],
   ["1997", 1353, 0.1],
   ["1998", 1142, 0.2],
   ["1999", 1201, 0.2],
   ["2001", 1201, 0.1]])
"1996"  1242  2/5
"1997"  1353  0.1
"1998"  1142  0.2
"1999"  1201  0.2
"2001"  1201  0.1

We use the data in the first and third column:

f := stats::empiricalPF(s, 2):
f(1242), f(1353), f(1200), f(1201)

delete s, f:

Example 3

We consider a fair die:

f:= stats::empiricalPF([1, 2, 3, 4, 5, 6]):

What is the probabiliy that tossing the die produces a score more than or equal to 4?

f({4, 5, 6})

delete f:

Example 4

Duplicate data elements are automatically combined to a single data element, adding up the corresponding probability values:

f:= stats::empiricalPF([1, 2, 1, 1, 2]):
f(1), f(2)

delete f:

Parameters

x1, x2, …

The statistical data: real numerical values

s

A sample of domain type stats::sample

c

A column index of the sample s: a positive integer. This column provides the data x1, x2 etc. There is no need to specify a column number c if the sample has only one non-string column.

Return Values

procedure.

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