Given $X\sim N(\mu_X, \sigma_X^2)$ and $Y\sim N(\mu_Y, \sigma_Y^2)$ are independent, and you know $X+Y=s$. What is the expected value of $X$?
I encountered this during an interview. My thoughts were to use conditional expectation $E(X|Z=s)$ where $Z\sim N(\mu_X+\mu_Y, \sigma_X^2+\sigma_Y^2)$. However, it involves a lot of calculations, which I don't think would be an interview question. Could anyone suggest?