Understanding the relationship between a distribution, an event, and a random variable I'm watching some MOOC cryptography videos and the lecturer is covering a bit of discrete probability theory. It's been a while since I covered any probability and this is a bit more advanced than what I learned. (Stanford's "basic" discrete math is apparently more than the one I had) 
I want to make sure I get these concepts here, so my question is:
Q: Is the below a reasonable working understanding of the gist of a probability distribution, event, discrete random variable, and random algorithm? Some of the notation is a bit confusing.
I'm going to assume I made mistakes so any feedback is greatly appreciated.


*

*A probability distribution defines a function mapping events to a range $[0,1]$

*An event is a subset of the sample space (are the objects in the set considered equivalent?)

*The probability of anything from an event set occurring is the sum of the probabilities of its elements (naturally)

*The probability of a union of events is bounded above by the sum of their probabilities (union bound / Boole's inequality, seems straightforward)

*A discrete random variable is a function mapping a value from $U$ to some other set $V$ (it is said to "take values in V" -- why "take"?) and it defines a distribution on $V$

*$Pr[X=x]$ represents the probability that event x will occur under the probability distribution defined by random variable (function) $X$ (I'm especially interested in getting this right since he seems to use this syntax a lot)

*A uniform random variable is a random variable whose probabilities are all $1/|U|$ (simple enough)

*The notation for a uniform random variable is $r \xleftarrow R U$ -- what does the $R$ represent here?

*A randomized algorithm has an implicit parameter $r$ (a random variable).

*A randomized algorithm points to a region of outputs (this matches with Shannon's "region of uncertainty" for signals in n-space, so that makes sense)

*The output of a randomized algorithm is a random variable taken from the codomain, representing a probability distribution over a subset of the codomain. (I think?)


Thanks.
 A: 
  
*
  
*A probability distribution defines a function mapping events to a range $[0,1]$
  

Yes, the function maps subsets of the sample space (events) to that real interval.   Further it is required that the probability measure for the entire sample space equals $1$, and the probability measure for the empty set equals zero.

  
*
  
*An event is a set of (equivalent?) objects
  

An event is a subset of the sample space; a set of outcomes.

  
*
  
*The probability of anything from an event set occurring is the sum of the probabilities of its elements (naturally)
  

Well, that works for sample spaces of countable many outcomes.
More generally, the probability for any union of pairwise disjoint events is the sum of the probabilities for each event.
$$\forall A\subseteq U~\forall B\subseteq U: \Big(~A\cap B=\emptyset \to \big(\mathsf P(A\cup B)=\mathsf P(A)+\mathsf P(B)\big)~\Big)$$

  
*
  
*The probability of a union of events is bounded above by the sum of their probabilities (union bound / Boole's inequality, seems straightforward)
  

Yes.   Even when such events may not be disjoint, the probability for their union cannot be greater than the sum of their probabilities. 
$$\mathsf P(\bigcup_i E_i) \leq \sum_i \mathsf P(E_i)$$

  
*
  
*A discrete random variable is a function mapping a value from $U$ to some other set $V$ (it is said to "take values in V" -- why "take"?) and it defines a distribution on $V$
  

Well, if $U$ is the sample space, $V$ some measure space, and the image only contains countably many values, then that is so.   Outcomes are said to 'take values', in the sense that they are assigned them by this mapping.

  
*
  
*$Pr[X=x]$ represents the probability that event x will occur under the probability distribution defined by random variable (function) $X$ (I'm especially interested in getting this right since he seems to use this syntax a lot)
  

Then we'll take it slow.


*

*$X$ is a discrete random variable.   It is a function of the sample space ($U$) mapping to measure space $V$.   It is to be understood that $X$ on its own stands for: the measure of the outcome which is realised (ie: that "happens") .

*$x$ is a value in that measure space. 

*$X=x$ is an event.
It is shorthand for the event $\{\omega\in U: X(\omega)=x\}$, the set of all outcomes taking value $x$ (ie: all outcomes with an $X$-measure of $x$).

*And so $\Pr(X=x)$ is the probability for that event.

  
*
  
*A uniform random variable is a random variable whose probabilities are all $1/|U|$ (simple enough)
  

That is for a uniform discrete random variable.   Later you will meet uniform continuous random variables which have a similar definition.

  
*
  
*The notation for a uniform random variable is $r \xleftarrow R U$ -- what does the $R$ represent here?
  

"Selected at Random", presumably.   I am not familiar with this notation.
