Chapter 4.5, Problem 11E

For a female freshman in a health fitness program, let X equal her percentage of body fat at the beginning of the program and Y equal the change in her percentage of body fat measured at the end of the program. Assume that X and Y have a bivariate normal distribution with ${\mu }_X=24.5,{\sigma }_X^2={4.8}^2=23.04,{\mu }_Y=-0.2,{\sigma }_Y^2={3.0}^2=9.0$, and $\rho =-0.32$.

Find

(a) $P\left(1.3\le Y\le 5.8\right)$.

(b) ${\mu }_{Y|x}$, the conditional mean of Y, given that $X=x$.

(c) ${\sigma }_{Y|x}^2$, the conditional variance of Y, given that $X=x$.

(d) $P\left(1.3\le Y\le 5.8|X=18\right)$.