Uniform Distribution

A distribution which has constant probability is called a uniform distribution, sometimes also called a Rectangular Distribution. The probability density function and cumulative distribution function for a continuous uniform distribution are

$$P(x) = \left\{\begin{array}{ll} {1\over b-a} & \mbox{for $a<x<b$}\\ 0 & \mbox{for $x<a$, $x>b$}\end{array}\right.$$ (1)

$$D(x) = \left\{\begin{array}{ll} 0 & \mbox{for $x < a$}\\ {x-a\over b-a} & \mbox{for $a \leq x < b$}\\ 1 & \mbox{for $x \geq b$.}\end{array}\right.$$ (2)

With $a=0$ and $b=1$, these can be written

$$P(x) = {\textstyle{1\over 2}}\mathop{\rm sgn}\nolimits (x)-\mathop{\rm sgn}\nolimits (x-1)$$ (3)

$$D(x) = {\textstyle{1\over 2}}[1-(1-x)^2\mathop{\rm sgn}\nolimits (1-x)+x\mathop{\rm sgn}\nolimits (x)].$$ (4)

The Characteristic Function is

$$\phi(t)={2\over ht}\sin({\textstyle{1\over 2}}ht)e^{imt},$$ (5)

where

$$a = m-{\textstyle{1\over 2}}h$$ (6)

$$b = m+{\textstyle{1\over 2}}h.$$ (7)

The Moment-Generating Function is

$$M(t)=\left\langle{e^{xt}}\right\rangle{} =\int_a^b {e^{xt}\over b-a} dx=\left[{e^{xt}\over t(b-a)}\right]_a^b,$$ (8)

so

$$M(t) = \cases{ {e^{tb}-e^{ta}\over t(b-a)} & for $t \not = 0$\cr 0 & for $t = 0$,\cr}$$ (9)

and

$$M'(t) = {1\over b-a}\left[{{1\over t}(be^{bt}-ae^{at})-{1\over t^2}(e^{bt}-e^{at})}\right]$$

$$= {e^{bt}(bt-1)-e^{at}(at-1)\over (b-a)t^2}.$$ (10)

The function is not differentiable at zero, so the Moments cannot be found using the standard technique. They can, however, be found by direct integration. The Moments about 0 are

$$\mu_1' = {\textstyle{1\over 2}}(a+b)$$ (11)

$$\mu_2' = {\textstyle{1\over 3}}(a^2+ab+b^2)$$ (12)

$$\mu_3' = {\textstyle{1\over 4}}(a+b)(a^2+b^2)$$ (13)

$$\mu_4' = {\textstyle{1\over 5}}(a^4+a^3b+a^2b^2+ab^3+b^4).$$ (14)

The Moments about the Mean are

$$\mu_1 = 0$$ (15)

$$\mu_2 = {\textstyle{1\over 12}}(b-a)^2$$ (16)

$$\mu_3 = 0$$ (17)

$$\mu_4 = {\textstyle{1\over 80}}(b-a)^4,$$ (18)

so the Mean, Variance, Skewness, and Kurtosis are

$$\mu = {\textstyle{1\over 2}}(a+b)$$ (19)

$$\sigma^2 = \mu_2={\textstyle{1\over 12}}(b-a)^2$$ (20)

$$\gamma_1 = {\mu_3\over\sigma^{3/2}}=0$$ (21)

$$\gamma_2 = -{\textstyle{6\over 5}}.$$ (22)

The probability distribution function and cumulative distributions function for a discrete uniform distribution are

$$P(n) = {1\over N}$$ (23)

$$D(n) = {n\over N}$$ (24)

for $n=1$, ..., $N$. The Moment-Generating Function is

$$M(t) = \left\langle{e^{nt}}\right\rangle{}=\sum_{n=1}^N {1\over N} e^{nt} = {1\over N} {e^t-e^{t(N+1)}\over 1-e^t}$$

$$= {e^t(1-e^{Nt})\over N(1-e^t)}.$$ (25)

The Moments about 0 are

$$\mu'_m={1\over N} \sum_{n=1}^N n^m,$$ (26)

so

$$\mu_1' = {\textstyle{1\over 2}}(N+1)$$ (27)

$$\mu_2' = {\textstyle{1\over 6}}(N+1)(2N+1)$$ (28)

$$\mu_3' = {\textstyle{1\over 4}}N(N+1)^2$$ (29)

$$\mu_4' = {\textstyle{1\over 30}}(N+1)(2N+1)(3N^2+3N-1),$$ (30)

and the Moments about the Mean are

$$\mu_2 = {\textstyle{1\over 12}}(N-1)(N+1)$$ (31)

$$\mu_3 = 0$$ (32)

$$\mu_4 = {\textstyle{1\over 240}}(N-1)(N+1)(3N^2-7).$$ (33)

The Mean, Variance, Skewness, and Kurtosis are

$$\mu = {\textstyle{1\over 2}}(N+1)$$ (34)

$$\sigma^2 = \mu_2={\textstyle{1\over 12}}(N-1)(N+1)$$ (35)

$$\gamma_1 = {\mu_3\over\sigma^{3/2}}=0$$ (36)

$$\gamma_2 = {6(N^2+1)\over 5(N-1)(N+1)}.$$ (37)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 531 and 533, 1987.