Conditional Expectation

• Adam’s Law: $E(E(Y\mid X))=E(Y)$
• Projection interpretation: $Y-E(Y\mid X)$, which is called the residual from using $X$ to predict $Y$, is uncorrelated with $h(X)$ for any function $h$.

Examples

Time until HH vs. HT

• Let $W_{HT}$ be the number of tosses until $HT$ appears.
  • $W_{HT}$ can be divided into time waiting for the first $H$ and that for the first $T$, hence $W_1$ and $W_2$.
  • Note that our partial progress is not destroyed. Hence $W_1,W_2\stackrel{\text{i.i.d}}{\sim }\text{FS}(1/2)$, and $W_{HT}=W_1+W_2$.
• The case for $W_{HH}$ is more complex, as our partial progress is destroyed.
  • Use the law of total expectation.
  • Condition on the first toss.
  • Use memorylessness.
  • Condition on the second toss.