## 5. Heights of Mothers and Daughters ##

The heights of a population of mother-daughter pairs have a bivariate normal distribution with correlation 0.5.

**a)** Of the mothers on the 90th percentile of mothers' heights, what proportion have daughters who are taller than the 90th percentile of daughters' heights?

**b)** In what proportion of mother-daughter pairs are both women taller than average? (This means the mothers are taller than the average mother and the daughters are taller than the average daughter.)

[Hint: Express standard bivariate normal variables in terms of two independent standard normal variables, and then apply the "slices of a normal cake" method.]