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I came across the following notation that I cannot follow: $1_{[0,1/2]}$ It is supposed to be some kind of random variable (or just an event? not sure)

It is hard to google this, too. What does such a random variable mean? If it helps, it was defined given a probability space $([0,1],B(0,1),L)$ where $B(0,1)$ contains Borel sets intersecting $[0,1]$ and $L$ is the Lebesgue measure.

Is there a general way to google specific notations that I might miss? I have skimmed through the wiki articles of Borel sets and the Lebesgue measure without success.

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    $\begingroup$ If I had to guess, I would say "uniform distribution on $[0, 1/2]$", but I dunno. $\endgroup$
    – Arthur
    Commented Jul 20, 2016 at 9:35
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    $\begingroup$ $\mathbf{1}_\mathbf{A}$ for a set $A$ often designates the characteristic function of $\mathbf{A}$, here $\mathbf{A}=[0,\frac{1}{2}]$. When $\mathbf{A}$ is Borel (which is obviously the case here), $\mathbf{1}_\mathbf{A}$ is thus a random variable (of total measure $\mu(\mathbf{A})$). $\endgroup$
    – yago
    Commented Jul 20, 2016 at 9:39
  • $\begingroup$ @YannHamdaoui: Please make that an answer. $\endgroup$ Commented Jul 20, 2016 at 9:44

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For a set $\mathbf{A}$, $\mathbf{1}_\mathbf{A}$ often designates the characteristic function of $\mathbf{A}$, that is, the function defined by $\mathbf{1}_\mathbf{A}(x) = 1 \text { if } x \in \mathbf{A}$ and $\mathbf{1}_\mathbf{A}(x) = 0$ otherwise.

When $\mathbf{A}$ is measurable, so is $\mathbf{1}_\mathbf{A}$ and $\int \mathbf{1}_\mathbf{A}d\mu = \mu(\mathbf{A})$

In your case, $\mathbf{A} = [0, \frac{1}{2}]$ and $\mu$ is the Lebesgue measure, so $\mathbf{1}_{[0,\frac{1}{2}]}$ is a random variable of expectation $\frac{1}{2}$.

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  • $\begingroup$ Thank you very much! So, is $[0,\frac{1}{2}]$ an open interval in our case? If yes, what exactly is the meaning of this interval? If it reflects the values of the cumulative distribution function, this would make sense (and then, is there a linear slope for values in between or is it just not defined?) If this can be answered by just studying Borel sets and Lebesgue's measure more thoroughly, let me know $\endgroup$
    – IceFire
    Commented Jul 20, 2016 at 9:59
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    $\begingroup$ You're welcome. $[0,\frac{1}{2}]$ is the closed interval containing all $x$ between $0$ and $\frac{1}{2}$ included, id est the set $\{ x \in \mathbb{R} \mid 0 \leq x \leq \frac{1}{2} \}$. Open intervals (containing all $x$ bewteen two reals but excluding these reals) are written with parenthesis like $(0,\frac{1}{2})$ for example, or with reverse brackets as $]0,\frac{1}{2}[$. $\endgroup$
    – yago
    Commented Jul 20, 2016 at 10:06
  • $\begingroup$ thank you! Still I do not understand exactly what the meaning of the interval in this context is. I see we can integrate it to get the mean... Not sure if there is more to it, I just don't see the value of writing it this way. Is this another notation to write that the values of this variable are in the range $[0, \frac{1}{2}]$ and uniformly distributed? $\endgroup$
    – IceFire
    Commented Jul 20, 2016 at 14:48
  • $\begingroup$ I guess this is indeed just a convenient notation but also widely used (not just in probability, but in set theory, real analysis, topology, etc.). As far as I can tell, I don't know another notation and this one would be the first one and the most natural to come to my mind if I had to refer to this function, no matter the context. $\endgroup$
    – yago
    Commented Jul 21, 2016 at 20:48
  • $\begingroup$ $\chi $ (to remind us of the word "characteristic) is sometimes used. $I$ (to remind us of the word indicator) is also sometimes used. The actual syntax is pretty universal though. Anyway no, it has nothing to do with the values: the values are 0 or 1, depending on whether the sampling variable $\omega $ is in $[0,1/2] $ or not. $\endgroup$
    – Ian
    Commented Jul 24, 2016 at 10:02

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