Joint distribution with independence testing

Questions that explicitly ask whether the random variables are independent or state that they are independent and use this property to find unknown probabilities.

2 questions

OCR S4 2012 June Q6
6 The random variables \(S\) and \(T\) are independent and have joint probability distribution given in the table.
\(S\)
\cline { 2 - 5 }012
\cline { 2 - 5 }1\(a\)0.18\(b\)
20.080.120.20
\cline { 2 - 5 }
\cline { 2 - 5 }
  1. Show that \(a = 0.12\) and find the value of \(b\).
  2. Find \(\mathrm { P } ( T - S = 1 )\).
  3. Find \(\operatorname { Var } ( T - S )\).
OCR S4 2009 June Q5
5 Alana and Ben work for an estate agent. The joint probability distribution of the number of houses they sell in a randomly chosen week, \(X _ { A }\) and \(X _ { B }\) respectively, is shown in the table.
\includegraphics[max width=\textwidth, alt={}, center]{f1879b0f-17e3-41b4-af38-a843b67c5301-3_405_602_370_781}
  1. Find \(\mathrm { E } \left( X _ { A } \right)\) and \(\operatorname { Var } \left( X _ { A } \right)\).
  2. Determine whether \(X _ { A }\) and \(X _ { B }\) are independent.
  3. Given that \(\mathrm { E } \left( X _ { B } \right) = 1.15 , \operatorname { Var } \left( X _ { B } \right) = 0.8275\) and \(\mathrm { E } \left( X _ { A } X _ { B } \right) = 1.09\), find \(\operatorname { Cov } \left( X _ { A } , X _ { B } \right)\) and \(\operatorname { Var } \left( X _ { A } - X _ { B } \right)\).
  4. During a particular week only one house was sold by Alana and Ben. Find the probability that it was sold by Alana.