I have two normally distributed random variables, X and Y.
X has mean 66 and standard deviation 6.
Y has mean 77 and standard deviation 7.
The correlation between the random variables is given as 0.8.
The task is to find the probability $P(X > 0.7Y)$
This is my attempt:
Create the random variable $D = 0.7Y - X$
and calculate $P(D < 0)$
The variable D should also be normally distributed with these properties:
$\mu_D = 0.7 \times 77 - 66 = -12.1$ $\sigma^2_D = 0.7^2 \times 49 + 36 + 2 \times 0.7 \times -1 \times 0.8 \times 7 \times 6 = 12.97$
Then, $P(D < 0) = F_Z\left(\frac{0 + 12.11}{\sqrt{12.97}}\right) \approx F_Z(3.36)$
But, this is not the answer I'm supposed to get (which is instead $F_Z(1.17)$ according to the textbook). Where am I going wrong?