Question

and are independent Gaussian (Normal) random Variables. has mean 13.9 and variance 5.2; has mean 6.9 and variance 3.8. . (a) Calculate

Answer 1

This question is incomplete, the complete question is;

X and Y are independent Gaussian (Normal) random Variables. X has mean 13.9 and variance 5.2; Y has mean 6.9 and variance 3.8. . (a) Calculate P( W> 10)

Answer:

P( W> 10) is 0.1587

Explanation:

Given that;

X ⇒ N( 13.9, 5.2 )

Y ⇒ N( 6.9, 3.8 )

W = X - Y

Therefore

E(W) = E(X) - E(Y)

= 13.9 - 6.9 = 7

Var(W) = Var(X) + Var(Y) -2COV(X.Y)

[ COV(X,Y) = 0 because they are independent]

Var(W) = 5.2 + 3.8 + 0

= 9

Therefore

W ⇒ N( 7, 9 )

so

P( W > 10 )

= 1 - P( W ≤ 10 )

= 1 - P( W-7 /3 ≤ 10-7 /3 )

= 1 - P( Z ≤ 1 ) [ Z = W-7 / 3 ⇒ N(0, 1) ]

from Standard normal distribution table, P( Z ≤ 1 ) = 0.8413

so

1 - P( Z ≤ 1 ) = 1 - 0.8413 = 0.1587

Therefore P( W> 10) is 0.1587