4.1.9 problem 9

Let $X$ be the amount of money Fred walks away with.

(a)

$\text{E}\left(X\right)=16000$. There is no variance under this scenario, since Fred’s take home amout is fixed.

(b)

$\text{E}\left(X\right)=\frac{1}{2}1000+\frac{1}{2}\frac{3}{4}32000+\frac{1}{2}\frac{1}{4}64000=20500.\text{Var}X=\text{E}\left(X^2\right)-{\left(\text{E}\right(X\left)\right)}^2\approx 4.76\ast 10^8.$

(c)

$\text{E}\left(X\right)=\frac{3}{4}1000+\frac{1}{4}\frac{1}{2}32000+\frac{1}{4}\frac{1}{2}64000=12750.\text{Var}X=\text{E}\left(X^2\right)-{\left(\text{E}\right(X\left)\right)}^2\approx 4.78\ast 10^8.$

Option $b$ has a higher expected win than option $c$, but it also has a higher variance.