Following up on this question, how would you derive the expectation and variance of the sum of two normally distributed random variables that aren't necessarily independent?
For example, if $$X \sim N(\mu, 3\sigma^2)$$ and $$Y \sim N(\mu + 9, \sigma^2)$$ is there a way to calculate the expectation and variance? What would the resulting X + Y distribution be in concrete terms? We aren't given the covariance.
Best Answer
$$E[X+Y]=E[X]+E[Y]=\mu+\mu+9=2 \mu +9$$ $$\begin{array} VVar[X+Y]&=Var[X]+Var[Y]+2Cov[X,Y]\\ &=3 \sigma^2+\sigma^2+E[XY]-E[X]E[Y]\\ &=4 \sigma^2+E[XY]-\mu(\mu+9) \end{array}$$
We can't say that much about the distribution of $XY$ without some more assumptions. If we know that $X$ and $Y$ are independent then $XY$ has a chi-squared distribution and we can compute $E[XY]$.