Solution Found!
Let X Mult5(n, p). (a) Find E(X1
Chapter 9, Problem 19(choose chapter or problem)
Let \(\mathbf{X}\sim \rm{Mult}_5(n,\ \mathbf{p})\).
(a) Find \(E\left(X_{1} \mid X_{2}\right)\) and \(\operatorname{Var}\left(X_{1} \mid X_{2}\right)\).
(b) Find \(E\left(X_{1} \mid X_{2}+X_{3}\right)\).
Questions & Answers
QUESTION:
Let \(\mathbf{X}\sim \rm{Mult}_5(n,\ \mathbf{p})\).
(a) Find \(E\left(X_{1} \mid X_{2}\right)\) and \(\operatorname{Var}\left(X_{1} \mid X_{2}\right)\).
(b) Find \(E\left(X_{1} \mid X_{2}+X_{3}\right)\).
ANSWER:Step 1 of 3
(a)
If we are given that \(X_{2}=m\), observe that we now have four remaining components to consider. There exist \(n-m\) non-decided items to put in each box, and the probability for each box is
\(\tilde{p}_i=\frac{p_i}{1-p_2},\ i=1,\ 3,\ 4,\ 5\)
since we have to have that
\(1=\sum_{i} \tilde{p}_{i}=\sum_{i} \frac{p_{i}}{1-p_{1}}=\frac{\sum_{i} p_{i}}{1-p_{1}}=\frac{1-p_{1}}{1-p_{1}}=1\)