2018.09.07

$\DeclareMathOperator{\Var}{Var}$ $\DeclareMathOperator{\Cov}{Cov}$ $\DeclareMathOperator{\Span}{Span}$
  • Probability of two variables given in a table.

  • Conditional probability: $P(Y=y | X=x)$ means the probability that you get $y$ given that you know you got $x$.

    $$ P(Y=y | X=x) := \frac{P(Y=y \text{ and } X=x)}{P(X=x)}. $$

    Usually used like this:

    $$ P(Y=y \text{ and } X=x) = P(X=x) \cdot P(Y=y | X=x) .$$

  • Example when $E[X\cdot Y] \not= E[X]\cdot E[Y]$.

  • Independence. The events $X$ and $Y$ are independent if knowing $X$ gives you no information about $Y$. That is, $$P(Y=y | X=x) = P(Y=y) .$$

  • If $X$ and $Y$ are independent, then $E[X \cdot Y] = E[X] \cdot E[Y]$. Know the critical step where independence is used. The rest of the argument is not as important.

  • Example when $\Cov(X,Y)$ is not zero using numbers in a table.