StatsRunsCDF

StatsRunsCDF (n, r)

The StatsRunsCDF function returns the cumulative distribution function for the up and down runs distribution for total number of runs r in a random linear arrangement of n unequal elements. There is no closed form expression. It is computed numerically from the recursion of the probability density

$$\displaystyle f(r, n)=\frac{r f(r, n-1)+2 f(r-1, n-1)+(n-r) f(r-2, n-1)}{n},$$

with the intial condition

$$\displaystyle f(1, n)=\frac{2}{n!} .$$

References

Bradley, J.V., Distribution-Free Statistical Tests, Prentice Hall, Englewood Cliffs, New Jersey, 1968.

Olmstead, P.S., Distribution of sample arrangements for runs up and down, Annals of Mathematical Statistics, 17, 24-33, 1946..

See Also

Statistical Analysis, StatsSRTest