In probability and statistics, the Bates distribution, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval.^[1] This distribution is sometimes confused with the Irwin Hall distribution, which is the distribution of the sum (not mean) of n independent random variables uniformly distributed from 0 to 1.

Definition

The Bates distribution is the continuous probability distribution of the mean, X, of n independent uniformly distributed random variables on the unit interval, U_i:

X = \frac{1}{n}\sum_{k=1}^n U_k.

The equation defining the probability density function of a Bates distribution random variable x is

f_X(x;n)=\frac{n}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^k{n \choose k}\left(nx-k\right)^{n-1}\sgn(nx-k)

for x in the interval (0,1), and zero elsewhere. Here sgn(x − k) denotes the sign function:

\sgn\left(nx-k\right) = \begin{cases} -1 & nx < k \\ 0 & nx = k \\ 1 & nx > k. \end{cases}

More generally, the mean of n independent uniformly distributed random variables on the interval [a,b]

X_{(a,b)} = \frac{1}{n}\sum_{k=1}^n U_k(a,b).

would have the probability density function of

g(x;n,a,b) = f_X\left(\frac{x-a}{b-a};n\right) \text{ for } a \leq x \leq b \,

Notes

References

• Bates,G.E. (1955) "Joint distributions of time intervals for the occurence of succesive accidents in a generalized Polya urn scheme", Annals of Mathematical Statistics, 26, 705–720

fr:Loi Bates