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The definition of the probability density function of the continuous uniform distribution on Wikipedia is:

$$ f(x) = \left\{ \begin{array}{} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{for } x < a \text{ or } x > b \end{array} \right. $$

As usual, $\mathbb{P}(X=x) = 0$ for any $x \in [a,b]$. But this just mean that the it "almost surely" does not occurs, rather than it is "impossible".

However, when $x < a$ or $x > b$, $\mathbb{P}(X= x) = 0$ and in this case, it actually is impossible. Values in intervals such as $[c,d]$ where $c > b$ never ever occurs either, rather than just "almost surely" not occur.

I thought a nice way to resolve this would be to simply restrict the sample space to exclude outcomes that is impossible or never occurs. So the pdf would instead be:

$$f(x) = \frac{1}{b-a} \text{ for } a \leq x \leq b.$$

This way, a probability of $0$ always mean that the event almost surely does not occur, which (to me) feels intuitively satisfying.

I was wondering if this is already being done in measure theoretical probability theory? The definition of the sample space $\Omega$ that I have read always define it as the set of all possible outcomes.

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    $\begingroup$ There currently is a discussion of sample spaces going on at stats.stackexchange.com/questions/64167; much of it is relevant here. Note that you indeed can restrict the sample space to exclude outcomes that cannot occur. But that is separate from the definition of a probability distribution (or density) function of a random variable, which always is defined on the entire Real line. Sample spaces are almost never sets of Real numbers (except perhaps in math texts): they are sets of things in which we are interested, like people, experimental outcomes, or processes. $\endgroup$
    – whuber
    Jul 13, 2013 at 2:14
  • $\begingroup$ @whuber - thanks for the explanation and +1. $\endgroup$
    – Legendre
    Jul 13, 2013 at 8:27

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I think you are confusing the whole concept of probability density distribution. Any smooth continuous distribution has $P(X=x)=0$ for any $x$. This is the whole concept of probability distribution - its values don't have direct connection to probability, it represents a concentration of probability (hence the name of it). This is logically easy to get as there is infinite count of numbers between $a$ and $b$, so the probability of each of them is $1/\infty$ which is $0$. Makes perfect sense to me.

I don't see a problem with possible and impossible events as well. If $f(x)=0$ the event is impossible, otherwise possible. Your redefinition makes it only worse because now your distribution is undefined on some pieces - so, that makes the probability undefined as well...

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  • $\begingroup$ f(x) is not P(X=x). There is no contradiction. $\endgroup$
    – sashkello
    Jul 13, 2013 at 1:46
  • $\begingroup$ Thanks! $f(x)=0$ for impossible events basically resolved my issue! +1 and accepted. $\endgroup$
    – Legendre
    Jul 13, 2013 at 8:26

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