Joint distribution with covariance calculation

Questions that ask for covariance, correlation, or require computing E(XY) - E(X)E(Y) without focusing on independence.

3 questions

OCR S4 2007 June Q3
3 The discrete random variables \(X\) and \(Y\) have the joint probability distribution given in the following table.
\(X\)
\cline { 2 - 5 } \multicolumn{1}{l}{}- 101
10.240.220.04
20.260.180.06
  1. Show that \(\operatorname { Cov } ( X , Y ) = 0\).
  2. Find the conditional distribution of \(X\) given that \(Y = 2\).
OCR S4 2014 June Q5
5 Two discrete random variables \(X\) and \(Y\) have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 1 } { 27 }\).
  2. Find \(\mathrm { E } ( X )\).
  3. Find \(\operatorname { Cov } ( X , Y )\).
  4. Are \(X\) and \(Y\) independent? Give a reason for your answer.
  5. Find \(\mathrm { P } ( X = 1 \mid Y = 2 )\).
OCR S4 2016 June Q3
3 The table shows the joint probability distribution of two random variables \(X\) and \(Y\).
\cline { 2 - 5 } \multicolumn{2}{c|}{}\(Y\)
\cline { 2 - 5 } \multicolumn{2}{c|}{}012
\multirow{3}{*}{\(X\)}00.070.070.16
\cline { 2 - 5 }10.060.090.15
\cline { 2 - 5 }20.070.140.19
  1. Find \(\operatorname { Cov } ( X , Y )\).
  2. Are \(X\) and \(Y\) independent? Give a reason for your answer.
  3. Find \(\mathrm { P } ( X = 1 \mid X Y = 2 )\).