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There are a number of definitions. In mathematics, it depends on the context that the word is used.
One definition is for a "random event" where all outcomes are equally likely to occur. A random selection of lottery numbers means that in the long run, no one number is likely to occur more than another one. Random events are typically considered independent random events, in that one event can not affect the outcome of another one.
A "random variable" means that the probability of any value occurring corresponds to a given probability distribution function.
There is a common use of random, which means to act without a plan.
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I have included two links. A normal random variable is a random variable whose associated probability distribution is the normal probability distribution. By definition, a random variable has to have an associated distribution. The normal distribution (probability density function) is defined by a mathematical formula with a mean and standard deviation as parameters. The normal distribution is ofter called a bell-shaped curve, because of its symmetrical shape. It is not the only symmetrical distribution. The two links should provide more information beyond this simple definition.
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