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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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What do you know? What have you tried? Where are you stuck? (You could avoid giving orders and add the (homework) tag, see the FAQ. You could also reread your question: A=F?) –  Did Jul 7 '11 at 6:16
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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 $\emptyset$. –  Henry Jul 7 '11 at 7:04
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1 Answer

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,\})$ ?

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