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Define a discrete random variable. Let $(Ω, A, P )$ be a probability space with

$Ω = \{1,2,3,4,5,6\}$ and $F = \{Φ, \{1,3,5\}, \{2,4,6\}, Ω\}$.

Define functions $X$, $Y$, $Z$ on $Ω$ as $X(k)= k$, $Y(k) = 1$ or $0$ as $k$ is even or odd, and $Z(k) = k^2$ for $k$ belonging to $Ω$. Determine which of $X$, $Y$, or $Z$ are discrete random variables on the probability space.

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    The symbol for the empty set is based on the Scandinavian letter Ø, not the Greek letter Phi. In Latex `\emptyset` produces $\emptyset$.2011-07-07

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Didier Piau has already provided advice about how you could improve your post, please do as suggested.

About your question: You need to look up the definition of discrete random variable.

First of all, a (real) random variable needs to be a function from the probability space to (if it is a real one) $\mathbb{R}$. It is possible to use other codomains, you'd need to look up what you are supposed to use.

Second, since the probability space is a discrete space, all random variables are discrete random variables, so we can ignore the "discrete" here.

Third, a random variable needs to be measurable. In the case at hand, it is possible to check this for every provided example by hand. You need to check if $f^{-1} (A)$ for any measurable set $A$ in the codomain is an element of $F$.

Hint: What is $ Z^{-1} (\{1, 4,\})$ ?