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for example you toss a coin three times then let $X$ be the number of times you get heads.

so since the coin is tossed three times it means the sample space is $ \Omega = \{HHH,HHT,HTH,HTT,TTT,TTH,THT,THH\}$

with each sample point from the sample space above we can associate a number for X (either 3 or 2 or 1 or 0).

this means X is a function from $\Omega$ to the real numbers so It follows that X is a random variable.

but how can we prove that formally, I mean is there a more formal proof of it.

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    from the information given, you can't prove it's random, you would need some information about the distribution of potential outcomes in order to prove that, else it could be a heavily skewed distribution and a biased coin – 2017-02-26
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    To be honest that depends on your definition of random variable. Do you have one specifically or are you looking for one? – 2017-02-26
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    @ Cursed1701 if you don't mind I have one extra question : in the general case which definition would you use to show that it's a random variable ? – 2017-02-26
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    A random variable is simply a measurable function from the sample space to the rel numbers. In your case every function $\Omega\to\mathbb R$ is measurable, so there is really nothing to be done. – 2017-02-26
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    @Sentinel135 I've found more than one definition over the internet I would like to know the most popular and most used one in college – 2017-02-26
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    When people are first introduced to the concept it is usually before they have much experience (*if any at all*) with calculus and certainly before they've had any measure theory. As such, it is often informally defined in such a way that the layman can understand it. The true rigorous definitions are generally the ones relying on measure theory such as the definition found [here](https://en.wikipedia.org/wiki/Random_variable#Measure-theoretic_definition). – 2017-02-26
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    @123 that's why I was asking. It depends on the field of mathematics you're talking about too. There are many informal ones; a few formal in probability, depending on if you're dealing with discrete or continuous random variables; and one in measure theory. Also, the formal definition in measure theory is very general. Mariano's and JMoravitz definition is the one from measure theory. So long as there's a measure for $\Omega \to \mathbb R$ the function $\chi:\Omega \to \mathbb R$ exists, and $\chi$ is a random variable. – 2017-02-26

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