Uniform random variable difference with distribution

Question

Is uniform random variable same as the uniform distribution? If not, what is the difference? Can someone explain me the uniform distribution in simple words and give an example?

Answer 1

Technically speaking, there is no uniform random variable. There are random variables that follow a uniform distribution. However, they are often used interchangeably.

The uniform distribution may be discrete or continuous. In one dimension, it is always defined on a subset of $\mathbb{R}$. When the distribution is discrete, the uniform distribution assigns equal probabilities to all possible outcomes. For example, throwing a fair dice can be modelled with a discrete uniform distribution, in which all the possible outcomes $\lbrace1, 2, \dots, 6\rbrace$ have equal probability $1/6$.

In the continuous case, the uniform distribution is defined on $(a, b)\subseteq \mathbb{R}$, and its main characteristic is that the density function is constant for all $x \in (a,b)$, given by $f(x) = \frac{1}{b-a}.$ and that the probability of observations falling in subintervals with the same length is constant. Real-life examples of the continuous uniform distribution are not that common, but it is frequently used in simulation. A fun example is approximating $\pi$ with uniform distributions.