MULTINOM

Updated 2024-03-08 21:49:58.933000

Syntax

SELECT [westclintech].[wct].[MULTINOM] (
   <@x, int,>
  ,<@p, float,>)

Description

Use the aggregate function MULTINOM to calculate the probability distribution of the outcomes of a multinomial experiment. Use MULTINOM for statistical experiments where:

• The experiment consists of x repeated trials.• Each trial has a discrete number of possible outcomes.• On any given trial, the probability that a particular outcome will occur is constant• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

P=\frac{x!}{x_1!x_2!\ldots x_k!}p_1^{x_1}p_2^{x_2}\ldots p_k^{x_k}

Arguments

@p

The probability for that bin.

@x

The bin value for the trials.

Return Type

float

Remarks

If @p is NULL then @p = 1

If @x is NULL then it is not included in the calculation.

@p is internally normalized so that SUM(p) = 1

x! is the factorial of the sum of @x

Examples

Example #1

Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue. 5 marbles are randomly selected from the urn with replacement. What is the probability of selecting 2 green marbles and 3 blue marbles?

SELECT wct.MULTINOM(x, p) as MULTINOM

FROM

(

    VALUES

        ('red', 0, 2),

        ('green', 2, 3),

        ('blue', 3, 4)

) n (c, x, p);

This produces the following result.

{"columns":[{"field":"MULTINOM","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"MULTINOM":"0.0975461057765585"}]}

Example #2

Suppose we roll a dice 5 times and obtain three ones and two sixes; in other words, a 6-tuple of (3,0,0,0,0,2). We can calculate the probability with the following SQL.

SELECT wct.MULTINOM(x, p) as MULTINOM

FROM

(

    VALUES

        (3, NULL),

        (0, NULL),

        (0, NULL),

        (0, NULL),

        (0, NULL),

        (2, NULL)

) n (x, p);