# Understanding random variable notation

$$X$$ takes values in the set $$S$$, and $$Y$$ and $$Y'$$ are each uniformly distributed over set $$T$$

• Does "take values in the set" mean that $$X : S -> ??$$?
• Does "uniformly distributed over set $$T$$" mean $$Y: ?? -> T$$?
• What ?? be in both cases?

I find non formal notation very confusin

• "takes value in $S$" and "distributed over $T$" both mean that the values realised by the random variable belong to $S$ and $T$, respectively. The originating set is a probabilised set usually denoted $\Omega$, endowed with a $\sigma$ algebra $\mathcal A$ and a probability measure on $\mathcal A$. – Xi'an Feb 5 '19 at 13:14
• @Xi'an Unfortunately many authors use identical language to mean that $S$ is the domain of $X$! In light of this, absent more context, it seems impossible to determine what the quotation truly is intended to mean. – whuber Feb 5 '19 at 13:22
• @whuber personally speaking, I've never read anywhere "A takes values in B" to mean B is in the domain of A. It feels a rather odd way to define such relationship. – Lucas Farias Feb 5 '19 at 13:36

1)

No, it means $$X$$ assume values that belong to $$S$$.

2)

No, it means that the values of $$Y$$ belong to $$T$$, and there they are uniformly distributed.

3)

The correct definitions would be:

$$X: \Omega \to S \cup Y$$ and $$Y \in T$$, where $$\Omega$$ is a set of possible outcomes for the random variable $$X$$.

• in your point (1) does that mean that S is the sample space or the outcome? – ramborambo Feb 5 '19 at 14:19
• @ramborambo sample space, that's what 3) implies. – Lucas Farias Feb 5 '19 at 14:27