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Background:

When we consider a continuous random variable $X$ and consider some independent realisations of $X$, I am told that the realisations must be unique since $\text{Pr}(X=x)=0$. When we consider the long run frequency interpretation of probability, where $\text{Pr}(A)=\frac{\text{ Number of times you get event A}}{\text{Number of trials}}$, and we observe $x$ three times we have that $3/\text{Number of independent runs of the experiment}$ still will approach 0 as number of trials approaches infinity.

Question:

Why is it then impossible that I observere a finite amount of observations that are the same when I make a very large number of independent realisations of a RV? When we divide this number by the number of trials, the fraction will approach 0 as the number of trials approaches infinity, so that $\text{Pr}(X=x)=0$ still holds.

share|improve this question
    
@FredrikAa I'd encourage you to change your title to something more informative, e.g. "Why X=x is impossible for continuous random variables?" -- it would make your question easier to find for someone wondering about similar problem in the future. Btw, I provided an answer for your question in detail -- I think that it with the links provided answers your question, if not feel free to comment. – Tim 7 hours ago
    
Thanks for your comment and answer, @Tim. I will change the title. – FredrikAa 6 hours ago
    
But you won't observe X=3. You'll never observe X=3. You might observe X=3.00001229384873492064897456978136578913657891648975623894‌​76129837461289734618‌​9775625....... and so on. – immibis 5 hours ago
    
For "must be unique" it is better to say "will be unique with probability $1$" and for "impossible" better to say "zero probability". If you observe a value of a continuous random variable, the prior probability that you would observe that particular value was $0$ even though you did in fact observe it, so it was not "impossible" to observe – Henry 3 hours ago
up vote 7 down vote accepted

Actually nobody says that such event is impossible. Probability equal to zero is not the same as impossibility (check here, here and here). Probability that $X=x$ is equal to zero for continuous variables because chance of such event happening is infinitely small since there is an infinite number of real numbers.

Also from purely mathematical point of view informally $\tfrac{1}{\infty} = 0$ and more formally, since infinity is not a number, $\lim_{n\to\infty} \tfrac{1}{n}=0$, so saying that probability of something happening is infinitely small is literally the same as saying that it is equal to zero.

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Could you please explain the difference between impossible, implausible and improbable (the terms appear in your second link)? I am not that familiar with sigma-algebras and measure-theory. Should not every possible subset of the sample space be part of the sigma-algebra, so that if the event $A$ is part of the sample space then $A$ is in the sigma-algebra? – FredrikAa 6 hours ago
    
If I understand things correctly, the sample space consists of very possible outcome in the experiment. All outcomes that are not possible in the experiment are in the emptyset? Is this what we mean by saying $\text{Pr}(\emptyset)$=0 (since the emptyset does not contain any elements from $S$?). And $\text{Pr}(A)=0$ does not mean $A$ is impossible, since $A$ is a subset of the sample space? Is this correct, @Tim? – FredrikAa 6 hours ago
    
@FredrikAa Yes, you are correct. I do not think that the distinction between impossible and implausible makes much sense in practice, but as for impossible and improbable: throwing seven in six-sided die is impossible (set of all possible outcomes are numbers from 1 to 6). Throwing infinite number of heads in infinite number of throws while using fair coin is improbable, but if you consider all the combinations of all the possible chains of outcomes, then such is a member of such set. – Tim 6 hours ago
    
As $x$ is already used, I'd use another variable name in the limit $\lim_{t\to\infty} 1/t = 0$ to avoid confusion – JiK 5 hours ago
3  
From a "purely mathematical point of view" $1/\infty$ is either undefined (when working with a standard model of the reals) or else it is not zero when you are working with the nonstandard reals: it is a nonzero infinitesimal whose standard part is zero. – whuber♦ 4 hours ago

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