# Uniform Continuous Probability Distribution Family

## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Uniform_distribution_(continuous) Retrieved:2015-6-22.
• {{Probability distribution

|name = Uniform

|type = density

|pdf_image = PDF of the uniform probability distribution using the maximum convention at the transition points.
Using maximum convention|

|cdf_image = CDF of the uniform probability distribution.|

|notation = $\displaystyle{ \mathcal{U}(a, b) }$ or $\displaystyle{ \mathrm{unif}(a,b) }$ |parameters = $\displaystyle{ -\infty \lt a \lt b \lt \infty \, }$

|support = $\displaystyle{ x \in [a,b] }$ |pdf = $\displaystyle{ \begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases} }$ |cdf = $\displaystyle{ \begin{cases} \lt P\gt 0 & \text{for } x \lt a \\ \lt P\gt \frac{x-a}{b-a} & \text{for } x \in [a,b) \\ \lt P\gt 1 & \text{for } x \ge b \lt P\gt \end{cases} }$

|mean = $\displaystyle{ \tfrac{1}{2}(a+b) }$ |median = $\displaystyle{ \tfrac{1}{2}(a+b) }$ |mode = any value in $\displaystyle{ (a,b) }$ |variance = $\displaystyle{ \tfrac{1}{12}(b-a)^2 }$ |skewness = 0

|kurtosis = $\displaystyle{ -\tfrac{6}{5} }$ |entropy = $\displaystyle{ \ln(b-a) \, }$ |mgf = $\displaystyle{ \begin{cases} \frac{\mathrm{e}^{tb}-\mathrm{e}^{ta}}{t(b-a)} &\text{for } t \neq 0 \\ 1 &\text{for } t = 0 \end{cases} }$ |char = $\displaystyle{ \frac{\mathrm{e}^{itb}-\mathrm{e}^{ita}}{it(b-a)} }$ }}

In probability theory and statistics, the continuous uniform distribution or 'rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated U(a,b). It is the maximum entropy probability distribution for a random variate X under no constraint other than that it is contained in the distribution's support.