conflict of being random and having a pdf I am studying probability theory and have a question regarding it.
We say 'X' is a random variable and it follows some distribution say exponential for that matter.
My question is if it were really a RANDOM variable the how would one even have some kind of formula such as a pdf or pmf to find out the probabilities either by putting the values in the pmf or pdf or by integrating.
After defining the pdf doesn't it remain only a Variable and not RANDOM at all.
We can call that function to be function whose range is in [0,1] that's it .why is it called a pmf or pdf of a RANDOM VARIABLE.
I know it may sound fooloish and stupid to ask but i couldn't refrain from doing so.
Any explaination would be highly appreciated.
 A: Consider the random variable $X$ that has PDF $f_X(x) = 6x(1-x),$ for $0 < x < 1$
(and $0$ elsewhere. [That is, $X\sim Beta(2,2).$] It is possible to sample
random variables that have this distribution. Sampling a thousand of them,
I got the histogram below. The blue curve superimposed on the histogram is
the function $y = f_X(x).$ 
Going backwards from a thousand observations to find out the PDF is not so easy.
One attempt is the red curve, which is a found through a statistical
process called density estimation. 

If I sample a million observations,
the histogram will be very smooth and the two curves will be
almost coincident. If I use a hundred observations, the histogram
will typically look very uneven, and the density estimator will be very poor.

So, depending on your purpose and situation,
you might have several different viewpoints on what it means to be "random."
I suppose you are just starting a probability course now. As you move
along, you will have a chance to clarify your question and some of its answers.
