Chapter 4.5, Problem 7E

For a pair of gallinules, let X equal the weight in grams of the male and Y the weight in grams of the female. Assume that X and Y have a bivariate normal distribution with ${\mu }_X=415,{\sigma }_X^2=611,{\mu }_Y=347,{\sigma }_Y^2=689$, and $\rho =-0.25$.

Find

(a) $P\left(309.2<Y<380.6\right)$.

(b) $E\left(Y|x\right)$.

(c) $Var\left(Y|x\right)$.

(d) $P\left(309.2<Y<380.6|X=385.1\right)$.