Another understanding of a random variable is a function defined on the space of "events". It assigns probabilities to events, that's all. The word "random" means that you cannot tell ahead (predict) which event will occur. The function itself, called PDF (probability density function) for "extended" events, or just probability in case of discrete events (a dice) is called "uniform" if:

- has exactly the same value for each discrete event, or

- has the same value on equal length intervals, as Johannes has already written.

What is difference between continuous and discrete probability functions?

as we know that the probability function, say f, is a function from a random variable, say X, to the set of real number (R), if we take r.v. X as a discrete variable then f is called discrete probability function, while if we take X as a continuous variable, then f is said to be continuous probability function.

discrete: you have a countable (not necessarily finite) set of possible 'events', choices or states which may occur. Each of these has an associated probability. The sum of all probabilities must be one (one of the choices must be true, otherwise I did not enumerate all of the choices). The example of a dice is one with a finte set of results or 'states'. The number of excited quanta in a quantum harmonic oscillator is an example with discrete but infinitelely many 'choices' or states.

continuous: the random variable is defined as a set with finite measure such as e.g. a finite interval (subset of the set of real numbers). Example: 'exact' location in time at which a radioactive nucleus does emit an alpha particle. It can happen at *any* time (from 'now' on, if the decay did not happen until that 'now' moment).

For any specific moment T (T a real number > 'now') the probability cannot be defined to be finite. Finite probabilities are associated with intervals (of time in this case): the probability for the decay to happen in the interval (t,t+dt) is p(t).dt.

In the latter case, since the interval is semi-infinite, the distribution of probability p(t) cannot be uniform (and we know it's an exponential decay in this case).

Another understanding of a random variable is a function defined on the space of "events". It assigns probabilities to events, that's all. The word "random" means that you cannot tell ahead (predict) which event will occur. The function itself, called PDF (probability density function) for "extended" events, or just probability in case of discrete events (a dice) is called "uniform" if:

- has exactly the same value for each discrete event, or

- has the same value on equal length intervals, as Johannes has already written.

@Marek

>

that's not all, that's nonsense.

Since a random variable f is, as you correctly state, a function on the space X of events (often called the sample space) its values are simply numerical (let me assume that our 'random variable' is a real-valued function, f: X --> R) characteristics of the samples. The connection with probabilities is a bit more complex but very satisfying from a mathematical point of view: f allows 'to carry over' the probability measure P on X to a probability measure f(P) on f(X). If this transported measure is the normal measure ( which measures intervals by their length) then f is 'uniformly distributed'. The transported measure is, of course, defined as f(P)(s) := P(f⁻¹(s)).

@Ulrich:

From purely formal point of view you are perfectly right. Indeed, the value of random variable (it may be multidimensional!) is not the same as the probability of a given event, which necessarily has to be in range [0,1]. Nor it is the event itself. True.

Yet your comment only adds confusion as it raises a question like: what if your f⁻¹(s), i.e. the set of events with the value of random variable equal to f, in case of continuous event space, is disconnected? Even if it is simply connected, then there is a problem which measure (acting in space of events) is the one we think of when we consider whether or not the given random variable is uniformly distributed? Does it mean than the same random variable may be uniformly distributed for some of us but deprived of this feature for others? That would be a nonsense.

@Marek

Sorry that I probably disturbed you by writing f(R) instead of f(X) .

I have corrected this in my original post. Then everything is well defined and none of your argued ambiguities apply.

Recall

f⁻¹(s) := { x \in X | f(x) \in s }

I assumed s subset of f(X) to be clear from the context.

@Ulrich,

I understood correctly your earlier use of notation f⁻¹(s), no matter that it took me some time (my computer's display get dimmed while I was digesting your complete post - this happened for the first time to me);). Isn't your approach an overkill?

During my student's years I was told about funny accident happened to one mathematician. He invented some nice theory, with many correct theorems, all of them with elegant proofs. The problem was that the only space where his theory could be applicable appeared R^3, where all his theorems were already known and next to trivial.

While I appreciate your approach "ex post", i.e. when I already feel everything well ("understand") but there is a need to precisely define and order all facts, then starting from it (at entry level) - is a real pain.

Moreover, I still have a problem: which measure? And why just this one not the other?

There are many to choose from.

Avinash,

I'm puzzled what do you mean in the second part of your question. Various objects can be considered "the same" if: 1)there is an equivalence relation operating on the set the said objects belong to, and 2) they both are in the same equivalence class. More precisely, they are called then "equivalent", not "the same".

But you are asking this question in context of random variables. Are you asking when one may say that two random variables are the same in the sense of their identical distributions, or, maybe, you mean their expected values are "equal within given uncertainty"? Something else?

@Marek.

This is not my approach but the approach from Kolmogorov on of any mathematically decent introduction to probability theory.

Unfortunately there is so much semi-understood slang around that the jewels get easily hidden.

In my student years I was told that a young mathematician became the protege of an old statistics professor since he was so fluent in modern mathematical terminology.

Finally the young mathematician was made Dr., based on work that 'generalized' the body of classical statistics from R^n to arbitrary compact Banach spaces. He did not notice that there is only one compact Banach space. This space, no surprise to you, has a single element, the null vector.

An uniformly distributed random variable in a real interval is a variable such that, for any subinterval included in the interval, the probability to find the variable there is proportional to the lenth of the subinterval.

Example: an uniformly distributed random variable in [0, 1] (standard computer function) has a probability a to be found in [0, a].

@Ulrich:

I think we all here should be grateful to you for your difficult at first sight but nevertheless very elegant, and first of all mathematically correct, presentation. Me included.

Yet you are (deliberately?) avoiding touching the problem "which measure?". There is no single one, thus you may obtain various results when using different measures. This was probably the reason for de Finetti (great probabilist, too!) to say "probability doesn't exist". Maybe this is also why some other mathematicians look suspiciously at probabilist, even today. I'm not sure whether Kolmogorov's axioms of probability theory were already known when de Finetti made his statement.

But all this takes us too far from the original question, I think. Otherwise we should discuss the so called subjective probability as well.

 Let X be an outcome of an statistical experiment and let random variable X can take N different possible value. Then if the chance of every value of X to be appear as out came in the next trial are equal. Then X said to follow uniform distribution.