3 A box contains a large number of pea pods. The number of peas in a pod may be modelled by the random variable \(X\). The probability distribution of \(X\) is tabulated below.
| \(\boldsymbol { x }\) | 2 or fewer | 3 | 4 | 5 | 6 | 7 | 8 or more |
| \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\) | 0 | 0.1 | 0.2 | \(a\) | 0.3 | \(b\) | 0 |
- Two pods are picked randomly from the box. Find the probability that the number of peas in each pod is at most 4.
- It is given that \(\mathrm { E } ( X ) = 5.1\).
- Determine the values of \(a\) and \(b\).
- Hence show that \(\operatorname { Var } ( X ) = 1.29\).
- Some children play a game with the pods, randomly picking a pod and scoring points depending on the number of peas in the pod. For each pod picked, the number of points scored, \(N\), is found by doubling the number of peas in the pod and then subtracting 5.
Find the mean and the standard deviation of \(N\).
[0pt]
[3 marks]