Joint distribution with marginal/conditional probabilities

3 From the records of Mulcaster United Football Club the following distribution was suggested as a probability model for future matches. \(X\) and \(Y\) denoted the numbers of goals scored by the home team and the away team respectively. Use the model to find

1. \(\mathrm { E } ( X )\),
2. the probability that the away team wins a randomly chosen match,
3. the probability that the away team wins a randomly chosen match, given that the home team scores. One of the directors, an amateur statistician, finds that \(\operatorname { Cov } ( X , Y ) = 0.007\). He states that, as this value is very close to zero, \(X\) and \(Y\) may be considered to be independent.
4. Comment on the director's statement.

6 A City Council comprises 16 Labour members, 14 Conservative members and 6 members of Other parties. A sample of two members was chosen at random to represent the Council at an event. The number of Labour members and the number of Conservative members in this sample are denoted by \(L\) and \(C\) respectively. The joint probability distribution of \(L\) and \(C\) is given in the following table. \(C\)

\(L\)
	0	1	2
0	\(\frac { 1 } { 42 }\)	\(\frac { 16 } { 105 }\)	\(\frac { 4 } { 21 }\)
1	\(\frac { 2 } { 15 }\)	\(\frac { 16 } { 45 }\)	0
2	\(\frac { 13 } { 90 }\)	0	0

1. Verify the two non-zero probabilities in the table for which \(C = 1\).
2. Find the expected number of Conservatives in the sample.
3. Find the expected number of Other members in the sample.
4. Explain why \(L\) and \(C\) are not independent, and state what can be deduced about \(\operatorname { Cov } ( L , C )\).

1 An unbiased coin is tossed three times. The random variables \(F\) and \(S\) denote the total number of heads that occur in the first two tosses and the total number of heads that occur in the last two tosses respectively. The table above shows the joint probability distribution of \(F\) and \(S\).

1. Show how the entry \(\frac { 1 } { 4 }\) in the table is obtained.
2. Find \(\operatorname { Cov } ( F , S )\).

5 The numbers of daily morning operations, \(X\), and daily afternoon operations, \(Y\), in an operating theatre of a small private hospital can be modelled by the following bivariate probability distribution.

1. 1. State why \(\mathrm { E } ( X ) = 4\) and show that \(\operatorname { Var } ( X ) = 0.9\).
   2. Given that $$\mathrm { E } ( Y ) = 3.7 , \operatorname { Var } ( Y ) = 0.61 \text { and } \mathrm { E } ( X Y ) = 14.4$$ calculate values for \(\operatorname { Cov } ( X , Y )\) and \(\rho _ { X Y }\).
2. Calculate values for the mean and the variance of:
   1. \(T = X + Y\);
   2. \(\quad D = X - Y\).
      [0pt] [4 marks]