TITLE: Probability between two time's points
QUESTION [0 upvotes]: In the first page of the article "On some problems involving random number of random variables" which you can find in http://www.jstor.org/discover/10.2307/2239257?uid=3737952&uid=2&uid=4&sid=21104082186897, (see the image article_prob.jpg)
The equation 2.2 shows the probability of the time of observation (v) for an extreme value:
$P(E_v^t)=P\{\tau(v)\le t\}-P\{\tau(v+1)\le t\}$,
But as I show in the adjunted second adjunted figure , I think it will be expressed in this way:
$P(E_v^t)=P\{\tau(v+1)\le t\}-P\{\tau(v)\le t\}$,
Also, the probability can be the same but with different sign, i.e, if P(a)-P(b)= 10-7 = 3, P(b)-P(a) = 7-10=-3. But this isn't so logic under my point of view, since we expect an positive probability or null probability.
Is this an error of the author?, could you show me in a simple way with somw aritmetic operations or sets graphics (circles)?
Thanks you very much..

REPLY [0 votes]: As you correctly observe, $\tau(\nu)$ is a continuous random variable
whose probability density function can be graphed somewhat like the graph you drew.
In addition, $\tau(\nu + 1)$ is a continuous random variable, 
and the graph of its probability density looks somewhat like that of $\tau(\nu)$.
But $\tau(\nu)$ and $\tau(\nu + 1)$ are not identically distributed, so it is a mistake to try to graph $P(\tau(\nu) \le t)$ and $P(\tau(\nu + 1) \le t)$ as areas under the same curve.
Instead, you should simply ask what it means for the event $E_\nu^t$ to occur.  In fact, for any given $t$ and any sequence of the values $\tau(\nu)$, there are three possibilities:

$t < \tau(\nu)$
$\tau(\nu) \le t < \tau(\nu+1)$
$\tau(\nu+1) \le t$

Since $E_\nu^t$ is just the second outcome in this list, the probability of $E_\nu^t$ is the probability that neither of the other two events occur; and since those two events are disjoint,
$$P(E_\nu^t) = 1 - P(t < \tau(\nu)) - P(\tau(\nu+1) \le t).$$
Now consider the fact that $P(\tau(\nu) \le t) =1 - P(t < \tau(\nu))$, and you can see how the formula is derived.
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