Lecture - Probability MT22, XV

Are the basic formulas for discrete random variables like linearity of expectation, covariance, etc. also true for continuous random variables?

Yes.

What does it mean for continous random variables $X$ and $Y$ to be independent?

If $f _ {X, Y} (x, y) = f _ X(x) f _ Y(y)$ for all $x$ and $y$.

How can you rewrite $F _ {X,Y}(x,y)$ for independent continuous random variables $x$ and $y$?

\[F_{X,Y}(x, y) = F_X(x) F_Y(y)\]

If $X$ and $Y$ are jointly normally distributed random variables, what does $\text{Cov}(X, Y) = 0$ imply (and why do you need to be careful)?

\[\text{Cov(X, Y)} = 0 \iff X, Y \text{ indep.}\]

Need to be careful since this equivalance fails in general.

When you two random variables whose domain are $(0, 1)$, then what do probabilities correspond to?

Areas.