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I am reading "introduction to probability 2nd edition" by Joseph K. and Blitzstein. The state a definition of discrete random variables as follows:

A random variable X is said to be discrete if there is a finite list of values $a_1, a_2, . . . , a_n$ or an infinite list of values $a_1, a_2, . . .$ such that $P(X = a_j \text{ for some j}) = 1$.

P is the probability function. So with this definition, I am confused about what is meant by "such that $P(X = a_j \text{ for some j}) = 1$" because from my experience "for some j" usually means "for one specific j". But anyone can imagine an experiment with a sample space $S$ consisting of multiple events where no single event with have 100% chance of happening. So what is meant by this definition with respect to the probability function P?

Of course, my intuitive guess would be that they mean there must exist a certain number of events such that the accumulated probability of those is 100% - am I right and is the use of "for some" in the definition standard practice and thus there is in fact nothing wrong with it?

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    $\begingroup$ I agree the phrasing isn't ideal, but they just mean $P(X\not \in \{a_i\})=0$. $\endgroup$
    – lulu
    Oct 1 '19 at 14:38
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I agree that the wording isn't ideal, but it just means that among the possible values that the random variable, X, must take, the probability that X is equal to one of these values is 1.

X must take one of these discrete values.

This is in contrast to the continuous case, where this probability is 0.

Hope this helps.

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Noting that any list in the form of $$a_1,a_2,\cdots $$ is countable. Let $$A=\cup_{i\in \mathbb{N}}\{a_i\}$$Then $$P(X=a_i\text{ for some i})=1 \iff P(X\in A)=1\iff P(X\notin A)=0 $$ In this case, the total probability can be expressed as a sum$$1=\sum_{a\in A}P(X=a) $$ If it is impossible to give a countable set $A$ such that $P(X\in A)=1$, then we must be in the case that $$ \sum_{a\in A}P(X=a)<1 \qquad \text{for any countable }A $$ So the total probability will have to be expressed as an integral (or a mix of sum and integral), in which case we will be dealing with a continous density (as oppose to discrete masses) for continuous random variable (or a mix of continuous and discrete parts).

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