# Sets / Bijection / Countablity

Is the set finite, countably infinite, or uncountable?

a. The set of all real-valued random variables on a finite sample space.

b. The set of all integer-valued random variables defined on the sample space $W$ of positive integers, with $\operatorname{Pr}[w] = 1/(2^w)$

c. The set of all integer-valued random variables on a finite sample space.

d. The set of all possible functions from $\mathbb{Z}_{97}$ to $\mathbb{Z}_{97}$ (modulo 97).

e. $\mathbb{Z}^3 = \{(a,b,c): a,b,c \in \mathbb{Z}\}$ (the set of triples of integers)

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What do you know? What did you try? Why did this fail? – Did Dec 2 '11 at 7:14
A random variable is just a function. Already if the sample space has $1$ point, there are as many random variables on the sample space as there are real numbers, since the point can be mapped to any real. – André Nicolas Dec 2 '11 at 7:34

An $S$-valued random variable on a sample space $X$ is simply a function from $X$ to $S$. Thus, a real-valued random variable on a set $X$ is simply a function from $X$ to $\mathbb{R}$, and an integer-valued random variable on $X$ is a function from $X$ to $\mathbb{Z}$. Your questions (a) and (c) are therefore the same as the questions that I answered here. That answer should also help you with (d) and (e), but (d) shouldn’t cause any trouble anyway: $\mathbb{Z}_{97}$ has $97$ elements, and in $\mathbb{Z}_{97}\times\mathbb{Z}_{97}$ each of them appears in an ordered pair with each of them, so you have a total of $97^2$ elements, a finite number.
That leaves only (b). The information that $\text{Pr}[w]=1/2^w$ is superfluous: you’re just looking at functions from $W$ to $\mathbb{Z}$. There are at least as many of those as there are functions from $W$ to $\{0,1\}$, so ... ?