3.6 Computing by Hand
3.6.1 Compute the Expected Value
Let \(X\) represent the result of one roll of a 6 sided die where the events \(\omega \in \Omega\) are mapped using a straightforward function: \(X(\omega):\) is a function that counts the number of spots that are showing, and maps the number of dots to the corresponding integer, \(\mathbb{Z}\).
- Calculating by hand, what is the expected value \(X\), which we write as \(E[X]\)?
- After you have calculated \(E[X]\): Is it possible that the result of a roll is this value?
3.6.2 Playing a Gnome Game, Part 1
- Suppose that, out on a hike in the hills above campus, you happen across a gnome who asks you if you would like to play the following game:
- You pay the gnome a dollar, and guess a number between 0 and 6. So, let \(g \in \mathbb{R}: 0 \leq g \leq 6\).
- After you make your guess, the gnome rolls a dice, which comes up with a value \(d \in \mathbb{Z}: d \in \{1,2,3,4,5,6\}\).
- The gnome pays you \(p = 0.25 \times |d - g|\).
- First question: What is the best guess you can make?
- Second Question: Should you play this game?
Fill this in by hand.