4.1.7 problem 7

(a)

Let $R$ be the birthrank of the chosen child. Then,

$P(R=3)=\frac{20}{100}\frac{1}{3}=\frac{4}{60}$

$P(R=2)=\frac{50}{100}\frac{1}{2}+\frac{20}{100}\frac{1}{3}=\frac{19}{60}$

$P(R=1)=\frac{30}{100}+\frac{50}{100}\frac{1}{2}+\frac{20}{100}\frac{1}{3}=\frac{37}{60}$

$\text{E}\left(R\right)=1\frac{37}{60}+2\frac{19}{60}+3\frac{4}{60}=\frac{29}{20}$

$\text{Var}\left(R\right)=\text{E}\left(R^2\right)-{\left(\text{E(R)}\right)}^2=\frac{149}{60}-\frac{841}{400}\approx 0.38$

(b)

$\text{E}\left(R\right)=1\frac{100}{190}+2\frac{70}{190}+3\frac{20}{190}=\frac{30}{19}$

$\text{Var}\left(R\right)=\frac{56}{19}-{\left(\frac{30}{19}\right)}^2\approx 0.45$