2023 HCI P2 Q8

A biased \(4\)-sided die gives the scores \(1\), \(2\), \(3\), and \(4\) with the probabilities shown in the table, where \(a\) and \(b\) are constants. The random variable \(S\) denotes the score on the die.

It is given that the mean of the score is \(2.56\).

1. Show that \(b=\frac{7}{25}\).[2]
2. \({{S}_{1}}\) and \({{S}_{2}}\) are two independent observations of \(S\).
   1. Find the probability that the sum of the two independent observations of \(S\) is at least \(6\) given that one of the observed scores is \(3\),[3]
      Let \(Y=\left| {{S}_{1}}-{{S}_{2}} \right|\). Without finding the probability distribution table of \(Y\),
   2. find \(\mathrm{Var}\left( 2S-\mathrm{E}\left( Y \right) \right)\).[3]
   3. state the value of \(\mathrm{E}\left( Y-\mathrm{E}\left( Y \right) \right)\).[1]