4.9 Joint Distribution Practice

4.9.1 Professorial Mistakes (Discrete RVs)

• Let the number of questions that students ask be a RV, \(X\).
  

• Let \(X\) take on values: \(\{1, 2, 3\}\), each with probability \(1/3\).
  

• Every time a student asks a question, the instructor answers incorrectly with probability \(1/4\), independently of other questions.

• Let the RV \(Y\) be number of incorrect responses.

• Questions:

  • Compute the expectation of \(Y\), conditional on \(X\), \(E[Y|X]\)
  • Using the law of iterated expectations, compute \(E[Y] = E\big[E[Y|X]\big]\).

4.9.2 Continuous BLP

• Recall the PDF that we worked with earlier to produce the $CEF[Y|X].

\[ f(x,y) = \begin{cases} 2, & 0 \leq x \leq 1, 0 \leq y \leq x \\ 0, & otherwise \end{cases} \]

Find the \(BLP\) for \(Y\) as a function of \(X\). What, if anything, do you notice about this \(BLP\) and the \(CEF\)?