Is a random variable really random or just a function?

[woundedtiger4]

lets say that X is some random variable that takes +1 if rational otherwise -1. at http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx in example 4, can we consider g(x) as a random variable because it's behaviour is same, right? is random variable really random or just function?
I found this very interesting notes on random variable:

http://web.simmons.edu/~benoit/Markov%20+%20Stochastic%20Readings%20%28Net%29/s05_03n.pdf

it says that a random variable
- is a function, not a variable
- is a specific mapping, and not random
So a random variable is neither random nor variable.

Actually, I want to ask that if it is not random and not variable then why is it called random variable? I guess that we call it random variable because we don't want to mix it up with variable that we use in ordinary calculus/engineering-calc, am I right?

Thanks in advance.

 

[Stephen Tashi]

So a random variable is neither random nor variable.

I agree.

Saying "X is a random variable" tells you that there is a certain probability distribution. For X to be a particular random variable, the probability distribution must be a particular probability distribution, not a "random" one.

However, the idea (or at least the notation) of "X" for a random variable allows complicated things to be expressed concisel.y For example if you draw a random sample from the probability distribution for X and square the value that you drew, you are sampling "a function of a random variable" given by Y = X^2. The notation Y = X^2 doesn't mean that the probability distribution for Y is the square of the probability distribution for X.

One reason for the terminology "random variable" is that it alerts people to the fact that we intend to speak of "functions of the random variable" using a notation that resembles functions of ordinary variables.

 

[woundedtiger4]

I agree.

Saying "X is a random variable" tells you that there is a certain probability distribution. For X to be a particular random variable, the probability distribution must be a particular probability distribution, not a "random" one.

However, the idea (or at least the notation) of "X" for a random variable allows complicated things to be expressed concisel.y For example if you draw a random sample from the probability distribution for X and square the value that you drew, you are sampling "a function of a random variable" given by Y = X^2. The notation Y = X^2 doesn't mean that the probability distribution for Y is the square of the probability distribution for X.

One reason for the terminology "random variable" is that it alerts people to the fact that we intend to speak of "functions of the random variable" using a notation that resembles functions of ordinary variables.

Thank you sir.
Can you please also help me with the first part of my question? i.e.

lets say that X is some random variable that takes +1 if rational otherwise -1. at http://tutorial.math.lamar.edu/Class.../TheLimit.aspx in example 4, can we consider g(x) as a random variable because it's behaviour is same, right?

 

[haruspex]

lets say that X is some random variable that takes +1 if rational otherwise -1. at http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx in example 4, can we consider g(x) as a random variable because it's behaviour is same, right?

Is that the right link? Don't see anything about RVs there.
What do you mean by an RV that "takes +1 if rational otherwise -1"? If what is rational?

it says that a random variable
- is a function, not a variable
- is a specific mapping, and not random
So a random variable is neither random nor variable.

When we write y = f(x), f is a function and y is a dependent variable. But we commonly write y = y(x), which is a 'pun'. In the same way, a 'random variable' is a function from some notional event space to an observation, but we can also think of it as a variable dependent on the event variable, and the event variable describes the random events.

 

[ImaLooser]

lets say that X is some random variable that takes +1 if rational otherwise -1. at http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx in example 4, can we consider g(x) as a random variable because it's behaviour is same, right? is random variable really random or just function?
I found this very interesting notes on random variable:

http://web.simmons.edu/~benoit/Markov%20+%20Stochastic%20Readings%20%28Net%29/s05_03n.pdf

it says that a random variable
- is a function, not a variable
- is a specific mapping, and not random
So a random variable is neither random nor variable.

Actually, I want to ask that if it is not random and not variable then why is it called random variable? I guess that we call it random variable because we don't want to mix it up with variable that we use in ordinary calculus/engineering-calc, am I right?

Thanks in advance.

That is correct. The random variable idea came first, the Kolmogorov axioms with the function definition came second. The point was to get randomness out of there, as it is hard to work with. The random variable remains an intuitively useful concept though, the idea being repeated trials that give unpredictable answers. The random variable notation is handy.

 

[woundedtiger4]

Is that the right link? Don't see anything about RVs there.
What do you mean by an RV that "takes +1 if rational otherwise -1"? If what is rational?

The link is not about RVs, it is just an ordinary calculus. (it is not about RVs).
Sorry for giving wrong example. Assume that X is a random variable that takes +1 on head & -1 on tail, i.e.
X={ +1 if head, -1 if tail

My question is: at http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx in example 5, is the behaviour/characteristics of H(t) are same as of X? Because H(t) is taking 0 if t<0 & 1 if t>=0, if H(t) & X satisfy some conditions (t<0 or t>=0, head or tail) then they produce/map some values specific values (0, 1, +1, -1). If both H(t) & X have same nature then why do we call a function (of ordinary calculus) as a random variable in probability?

 

[haruspex]

I believe I answered that in the second part of my post (#4), and Imaloser provided some history. Essentially, RVs are functions, but calling them RVs instead helps the reader understand the context. Note that calling them random functions would have been quite misleading.