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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?

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    $\begingroup$ They are clearly not the same; one is a random variable, the other is a probability distribution. A uniform(-distributed) random variable is a random variable that has a uniform distribution. $\endgroup$ Mar 5, 2020 at 3:48

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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.

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