The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is 175cm. The standard deviation is 6cm for women and 8cm for men. The correlation of height between husbands and wives is 0.5.
Let X be the height of a typical wife and Y the height of her husband.
(a) Show how Y can be represented as a sum of term which is a multiple of X and a term which is independent of X. Hence or otherwise:
(b) Given that a woman has height 168cm, find the expected height of her husband.
(c) Given that a woman has height 168cm, what is the probability that her husband is above average height?
(d) What is the probability that a randomly chosen man is taller than a randomly chosen woman?
(e) What is the probability that a randomly chosen man is taller than his wife?
Q1) I am not sure what they mean by asking to represent $Y$ as a sum of two terms? in (a)
Q2) I can find answers to b) and c) by finding conditional distributions however I feel they want to use the relationship in part a)
Q3) What kind distribution should I use for d) and e) ? How do you tackle this in general?
$X = 165+ 6 Z_1$ Where $Z_n$ are standard normal random variables.
$Y = 175 + A Z_1 + B Z_2$ a component the depends on X and a component that is independent from X
$A^2 + B^2 = 64$ the variance of Y
$6A = cov(X,Y)\\ \rho = \frac {cov(X,Y)}{\sigma_x\sigma_y} = 0.5\\ \frac {6A}{48} = 0.5\\ A = 4\\ B = 4 \sqrt 3\\ Y = 175 + 4 Z_1 + 4 \sqrt 3 Z_2\\ \frac 43 X = 220 + 4Z_1\\ Y = - 45 + \frac 43 X + 4 \sqrt 3 Z_2$
b) the wife is 3 cm above average. We would expect her husband to be 2 cm above average. or 177 cm
c) $Y = 177 + 4 \sqrt 3 Z_2\\ Y - \overline Y = 2 + 4 \sqrt 3 Z\\ P(Z) = \frac{-2}{4\sqrt 3}$
d) If we sample the a non married couple, there is no correlation in their height.
$M = 175 + 8 Z_1\\ W = 165 + 6 Z_2\\ M-W = 10 + 8Z_1 - 6Z_2$
the variance of the sum (or difference) of two independent random variables is the sum of the their variance. The standard deviation is the square root of the variance.
$M-W = 10 + 10Z$
What is the probability that our sampled variable is to the right of -1 standard deviation?
e) $Y - X = 10 + 2Z_1 + 4\sqrt3 Z_2\\ Y - X = 10 + \sqrt{52} Z\\ P(Z) > \frac {-10}{\sqrt52}$