# Is Uniform distribution [a,b] always symmetric?

I want to know whether any uniform distributed random variable is symmetric on any interval [a,b]. My thinking is it is symmetric on any interval [a,b]. i tried to think about a counter-example. But I didn't find any. Is there any?

I want to know this as I want to relate the uniform distribution to the location family, so that I can calculate ancillary statistics. Because if uniform distribution is symmetric on any interval, then the statistics based on order statistics are always ancillary.

• What's your definition of symmetry? – Glen_b Sep 18 '17 at 5:18
• – whuber Sep 18 '17 at 13:25

Yes, by definition of symmetric distribution "A probability distribution is said to be symmetric if and only if there exists a value $x_{0}$ such that $f(x_{0}-\delta )=f(x_{0}+\delta )$ for all real numbers where f is the probability density function if the distribution is continuous or the probability mass function if the distribution is discrete."
For the Uniform distribution $U[a,b]$ the probability density function is equal to $\frac{1}{b-a},a<x<b$ or $0,x<a, x>b$,therefore $f(x_{0}-\delta ) =f(x_{0}+\delta )=\frac{1}{b-a}$ for a uniform distribution ($a<x_0-\delta<x_0+\delta<b)$