Question

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) = ___________

Answer 1