Suppose that height Y and arm span X for U. S. women, both measured in cm, are normally distributed with means E(Yi) = 168, E(Xi) = 165, variances var(Yi) = 21, var(Xi) = 28, and covariance cov(Xi, Yi) = 20 for measurements on the same individual. For the purpose of this question, the variables are jointly normally distributed, and the values are independent for distinct individuals.
Part a: The correlation between height and arm span is .
Part b: The ‘albatross index’ is the difference Di = Xi − Yi between arm span and height. The mean of D is E(Di) = .
The standard deviation is sd(Di) =
.
Part c: Find the following probabilities for one individual: P(Di >9)= ;
P(Xi >Yi)= ;
P (Xi + Yi > 330) =
Part d: Consider now two specific unrelated individuals named i and j respectively. Compute the following probabilities:
P (Xi − Xj > 10) = ; P (Xi + Xj < 320) = ; P(|Xi −Yi|<10)= ; P (|Xi − Yj | < 10) = .