Merge pull request #63 from nathanchi/main

Update index.html

chapters/part3/independent_vars/index.html

@@ -73,8 +73,8 @@ <h2>Expectation of Products</h2>
  <b><i>Lemma: Product of Expectation for Independent Random Variables</i></b>:<br/>
  If two random variables $X$ and $Y$ are independent, the expectation of their product is the product of the individual expectations.
 \begin{align*}
- &E[X \cdot Y] = E[X] \cdot E[Y] && \text{ if and only if $X$ and $Y$ are independent}\\
+ &E[X \cdot Y] = E[X] \cdot E[Y] && \text{ if $X$ and $Y$ are independent (i.e. Cov(X,Y) = 0).}\\
  &E[g(X)h(Y)] = E[g(X)]E[h(Y)] && \text{ where $g$ and $h$ are functions} 
 \end{align*}
 Note that this assumes that $X$ and $Y$ are independent. Contrast this to the sum version of this rule (expectation of sum of random variables, is the sum of individual expectations) which does <b>not</b> require the random variables to be independent.
- </div>
+ </div>