# Dual normal distribution

Let random variable X has dual normal distribution with mean vector $\mu$ and

covariance matrix $\sum$

$$X= \left ( \begin{array}{ccc} x_1 \\ x_2 \end{array} \right )$$ $$\mu= \left ( \begin{array}{ccc} 3\\ 5\end{array} \right )$$ $$\sum= \left ( \begin{array}{ccc} 4 & -2\\ -2 &4 \end{array} \right )$$ Let $y = 2x_1 - 3x_2$ Find $P(y>4)$

I know how to find $P((x_1, x_2) \subset D )$, but i have no idea how to find $P(y>4)$

## 1 Answer

hint

condition on one of the variables: $$\mathbb{P}[y>4] = \int_{-\infty}^\infty \mathbb{P}[y > 4|x_1=t] f_{x_1}(t) dt...$$