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

Part a

0.825

Part b

17.12

Part c

(I) 0.4325

(ii) 0.659

(iii) 2.5E-4

Part d

(I) 0.0902

(ii) 0.902

(iii)0.146

(iv) 0.1265

Step-by-step explanation: Check attachment