# Random variables: How would you explain it to a beginner?

Different types of random variables: (discrete) Binomial, hypergeometric, geometric, Poisson (continuous) Uniform, normal, exponential

Random variables are very useful tools when solving simple and complex problems related to probability. They're used in diverse situations in many different forms, so how should you, for instance, describe in very general terms what they are to a student who is just starting to learn about mathematics?

Not really looking for a formal definition here, but more of a "here is how it's relevant" to your studies and your life kind of 101-deal. Something that even a middle school or high schooler could understand.

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"Let $X$ be the number of times you can a sum of $7$ when you throw three dice. Then the probability distribution of $X$ is given by $\Pr(X=0)=\text{whatever}$, $\Pr(X=1)=\text{whatever}$, $\Pr(X=2)=\text{whatever}$, $\Pr(X=3)=\text{whatever}$." Etc. Then $X$ is an example of a "random variable". For continuous distributions, speak of the probability that $X$ is between two numbers, rather than the probability that $X$ is equal to some number. In other words, I would not start by stating a precise definition of the concept of "random variable".

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Strongly agree. Students can learn to use (correctly!) random variables in sentences before being exposed to formal definitions. – André Nicolas Nov 22 '11 at 2:07

Suppose you have a space $\Omega$ where you have a probability defined. The possible outcomes of tossing a coin, for example. Now, imagine you are gambling... for each possible outcome it is defined the amount of money you will get (or loose). This is a random variable!!

The catch is that with a function $f: \Omega \to \mathbb{R}$, you can transport the probability defined for $\Omega$ to a probability in $\mathbb{R}$. So, if for instance you get $10$ bucks when you get heads, but looses $5$ for tails, then you can talk about the probability of loosing $5$ bucks when you toss the coin...

Notice that it does not make much sense to ask the expected "value" for heads or tails. If instead of heads and tails you get a coin with faces coloured green and blue, it is probably meaningless to say that in mean, the expected colour is cyan... On the other hand, if you are talking about loosing and gaining money, it makes sense to talk about the expected amount of money you will gain or loose. And this is the expectation.

It is important to emphasise that a "random variable" is NOT a function that gives randomly different values for the same "input". The amount you get for heads or tails is always the same for a fixed $f$! It is just a way to transport the probability in $\Omega$ to a probability in "amounts" (real numbers), so you can talk about expectation.

Usually one is not interested in the random variable itself... people talk about random variable when they just want to talk about the probability they induce in $\mathbb{R}$. This induced probability is the distribution of the random variable.

Two random variables $f$ and $g$ are independent, for example, when knowing or not the outcome of $f$ (in terms of events: $f \in A \subset \mathbb{R}$), makes no difference in determining the probability of $g$'s outcome.

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My example also shows that you would enjoy very much gambling with me!! :-) – André Caldas Nov 22 '11 at 2:57
Plus one because of the example :) – Leandro Nov 23 '11 at 2:08

It seems to me that an intuitive application where random variables play an important role may help motivate the concept.

Consider the manager of a customer support center who has to decide how many customer support personnel to hire to man the telephone lines. The number of support personnel is dependent on the number of calls that come in; a number that is likely to vary depending on the time of the day, day of the week etc.

Thus, it seems reasonable to assume that the number of calls in any given time period is uncertain with a range of plausible values (say, the number of calls to arrive at the center per minute can range anywhere between 5 to 10). One way to capture the above scenario is to let $N$ be the number of calls that arrive per unit time period.

In the above scenario, we would call $N$ a random variable as we do now know for sure the value we would observe apriori (i.e., we do not know how many calls would come per minute). Then, we can (depending on the situation) assume that $P(N=5) = 0.2$, $P(N=6) = 0.3$ and so on to capture our uncertainty.

The advantage to the above approach is that a student immediately appreciates the practical application and utility of the concept of random variables. The disadvantage, however, is that it requires a more elaborate explanation as the application needs to be sufficiently realistic.

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This is one example where one is talking about the distribution and not the random variable itself. Notice that there is even no mention of $N$'s domain of definition. It is difficult to have a realistic example because the set $\Omega$, where $N$ is defined is not "realistic"... – André Caldas Nov 22 '11 at 3:01
@Andre I thought we were talking about beginners and motivating them to see why random variables are important. We can always polish up the example as much as desired. – tards Nov 22 '11 at 3:04
Sorry, @tards... I was not complaining. Well, maybe a little... I think we are used to complicating things too much. In your example, you do exactly what the author of the question does. You are not at all out of scope. I just think that people do not realize they are not talking about a random variable... they are just talking about a distribution over the reals. I just find it too complicated to put random variables when you do not need them. If you do not need (or do not care about) the domain of definition, you do not need the random variable. – André Caldas Nov 22 '11 at 3:20

A random variable is another word for a function. Its input is the outcome of a random event, and it returns some aspect of the outcome.

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