4. A roulette wheel used in a U.S. casino has 38 slots, of which 18 are red, 18 are black, and 2 are green. A roulette wheel used in a French casino has 37 slots, of which 18 are red, 18 are black, and 1 is green. A ball is rolled around the wheel and ends up in one of the slots with equal probability. Suppose that a player bets on red. If a $1 bet is placed, the player wins $1 if the ball ends up in a red slot. (The player's $1 bet is returned.) If the ball ends up in a black or green slot, the player loses $1. Find the expected value of this game to the player in (a) The United States. (b) France. Translation: $X$ in the US has the pmf $f(1) = 18/38$ and $f(-1) = 20/38$. What is $E[X]$ in the US? $X$ in France has the pmf $f(1) = 18/37$ and $f(-1) = 19/37$. What is $E[X]$ in France?
5. Find the mean and variance for the following discrete distributions:
(a) $f(x) = \frac{1}{5}$, $x = 5, 10, 15, 20, 25$.
(b) $f(x) = \frac{1}{5}$, $x = 5$.
(c) $f(x) = \frac{x}{6}$, $x = 1, 2, 3$.
6. Given $E[X + 4] = 10$ and $E[(X + 4)^2] = 116$, determine (a) $Var(X + 4)$, (b) $E[X]$, and (c) $Var(X)$.
