Question

Height Y and arm span X are normally distributed with means E(Y_i) = 168, E(X_i) = 165, variances var(Y_i)=21, var(X_i)=28, and covariance cov(X_i ,Y_i)=20 for measurements on the same individual. The variables are normally distributed, and the values are independent for distinct individuals.

Find the following probabilities for one individual:

P(D_i > 9) =

P(X_i > Y_i) =

P(X_i + Y_i > 330) =

Consider now two specific unrelated individuals named i and j respectively. Compute the following probabilities:

P(X_i - X_j > 10) =

P(X_i + X_j < 320) =

P(|X_i – Y_i | < 10) =

P(|X_i – Y_j | < 10) =

Answer 1

Solution:

Given that:

Y_i : Denotes the height of ith individual.

X_i : Denotes the arm span of ith individual.

E(X_i) = 165 ,V(X_i) =28 X_i N (165,28)

E(Y_i) = 168 ,V(Y_i) =21 Y_i N (168,21)

Cov (X_i , Y_i) = 20

E(Di) = 7 , V(Di) = 293

Now,