Quick theory overview:

A function \(\mathbb{X} : \Omega \to \mathbb{R}\) is a Random Variable with respect to a sigma algebra \(\mathcal{F}\) if the inverse image of every set \((-\infty, x]\) is an event in \(\mathcal{F}\). Formally:

For every \(x \in \mathbb{R}\), the set \(\{ \omega \in \Omega \mid \mathbb{X}(\omega) \leq x \}\) must be in \(\mathcal{F}\).

Procedure:

• A sample space \(\Omega\) and a randomly generated sigma algebra \(\mathcal{F}\) are given.
• Several candidate functions are displayed. Click on all the functions you believe are valid Random Variables with respect to the given \(\mathcal{F}\).
• Note that there may be one or more correct answers.
• Click "Check Answer". The "Observations" panel will tell you if you are correct. If you select an invalid function, it will show you exactly why it fails the test.
• Press "New Problem" to generate a new sigma algebra and a new set of functions.