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.
conflict of being random and having a pdf
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probability
probability-theory
probability-distributions
soft-question
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1If, say, $f_X(x)$ is the probability density function of the random variable $X$ then $x$ and $X$ are two different things. It is not $X$ that we substitute in $f_X(x)$. So, I don't really understand your question. – 2017-01-31
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0The standard mathematical approach to describing randomness doesn't necessarily match our intuitive understanding of randomness -- at least not everybody's. Your question is very deep, and you may try to develop new approaches that will fit your views. The standard approach is just the one that has worked best so far for practical purposes. – 2017-01-31
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0My personal impression is that there is no randomness in the language of modern probability theory. It simply focuses on the notion of how an event is likely to occur (and this information is encoded in the notion of *probability law*). In this way, it avoids the question of what is random and leaves this question to people who attempt to model a random phenomenon or to interpret a probability model. – 2017-01-31
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0It isn't called pdf of a random variable in the sense we call functions R->R functions of a real variable. With a pdf we don't map a random variable to some value, we map a real number to a real number. We call it a pdf of a random variable because we pair it with a random variable in order model the probability of that variable taking some value. – 2017-01-31
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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.