Merge pull request #4794 from HaccerKat/combinatorics

Expected Products

content/4_Gold/Combinatorics.mdx

@@ -1107,6 +1107,19 @@ print(expected_ops)
 </PySection>
 </LanguageSection>
 
+## Expected Products
+
+Linearity of expectation deals with $E[X + Y]$, but what about $E[X \cdot Y]$?
+
+$E[X \cdot Y] = E[X] \cdot E[Y]$ if $X$ and $Y$ are independent from each other.
+We can reconsider the example of a fair 6-sided die to show that $E[X^2] \neq E[X]^2$.
+We know that $E[X] = \frac{7}{2}$, so $E[X]^2 = \frac{7}{2} \cdot \frac{7}{2} = \frac{49}{4}$.
+
+On the other hand,
+$$
+E[X^2] = \sum_x x^2 \cdot P(X = x) = \frac{1 + 4 + 9 + 16 + 25 + 36}{6} = \frac{91}{6}.
+$$
+
 ## Problems
 
 <Problems problems="general" />