Updated: 31 July 2015

Use NEGBINOM_DIST to calculate the probability mass function or the cumulative distributive function of the negative binomial distribution. The negative binomial distribution calculates the probability of k successes occurring before r failures in a sequence of Bernoulli trials.

SELECT [wct].[NEGBINOM_DIST](

<@Number_f, int,>

,<@Number_s, int,>

,<@Probability_s, float,>

,<@Cumulative, bit,>)

the number of failures. *@Number_f* must be of a type **int** or of type that intrinsically converts to **int**.

the number of successes. *@Number_s* must be of a type **int** or of type that intrinsically converts to **int**.

the probability of success. *@Probability_s* must be of a type **float** or of type that intrinsically converts to **float**.

a **bit** value identifying whether the probability mass function ('False') or the cumulative distribution function ('True') is to be returned. *@Cumulative* is of a type **bit** or a type that implicitly converts to **bit**.

float

· 0 = *@Number_f*

· 0 < *@Number_s*

· 0 = *@P* = 1

Stephen Curry of the Golden State Warriors makes 91.4% of his free throws. In a game in which he attempts 7 free throws what is the likelihood the he will miss 2?

SELECT

wct.NEGBINOM_DIST(

2 --@Number_f

,5 --@Number_s

,.914 --@Probability_s

,'False' --@Cumulative

) as NEGBINOM_DIST

This produces the following result

This is the probability that he will miss exactly 2 free throws. To calculate the possibility that we will miss no more than 2 three throws we would enter the following SQL.

SELECT

wct.NEGBINOM_DIST(

2 --@Number_f

,5 --@Number_s

,.914 --@Probability_s

,'True' --@Cumulative

) as NEGBINOM_DIST

This produces the following result.

The calculation of the probability mass function is closely related to the combinatorial function.

SELECT

wct.NEGBINOM_DIST(failures,trials-failures,p,'False') as NEGBINOM_DIST

,wct.COMBIN(trials-1,failures)*POWER(p,trials-failures)*POWER(1-p,failures) as COMBIN

FROM (VALUES (2,7,cast(0.914 as float)))n(failures,trials,p)

This produces the following result.

The cumulative distribution function is closely related to the Beta distribution.

SELECT

wct.NEGBINOM_DIST(failures,trials-failures,p,'True') as NEGBINOM_DIST

,wct.BETA_DIST(p,trials-failures,failures+1,'True',NULL,NULL) as BETA_DIST

FROM (VALUES (2,7,cast(0.914 as float)))n(failures,trials,p)

This produces the following result.