Theorem 6.3.5. Properties of the Continuous Uniform Probability Function.
For the Continuous Uniform Distribution over \(R = [a,b]\text{,}\) with a < b,
- \(f(x) = \frac{1}{b-a}\) satisfies the properties of a probability function over R = [a,b].
- \(\displaystyle \mu = \frac{a+b}{2}\)
- \(\displaystyle \sigma^2 = \frac{(b-a)^2}{12}\)
- \(\displaystyle \gamma_1 = 0\)
- \(\displaystyle \gamma_2 = \frac{9 \, {\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} {\left(a - b\right)}}{5 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)}^{2}}\)