Let $X= (X_1,X_2)$ be a random vector with bivariate normal distribution $X\sim N(\mu,\Sigma)$ such that $X_1$ and $X_2$ are positively correlated and we also have to: $P(X_1<1) = 0,84134 $, $P(X_2>6) = 0,02275 $, $Var[X_1] = 1 $ and $Var[X_2] = 2 $. How can I calculate the expected value $E[X_1\mid X_2= 6]$ and the covariance $Cov(X_1,X_2)$?.
I have tried to do the integral according to the expected value and Conditional Variance, but I have not been able to arrive at something concrete. Is there a miraculous property that I don't know about and that works for me?
We know $X_1$ and $X_2$ have variances $\sigma_1^2 = 1$ and $\sigma_2^2 = 2$ respectively. $0.84134 = P(Z < 1)$ and $0.02275 = P(Z > 2)$ for a standard normal random variable $Z$. That lets you figure out $\mu_1 = E[X_1]$ and $\mu_2 = E[X_2]$. But you have given us no information about the covariance $\Sigma_{12}$ except that it is positive.