7. A bag contains a large number of coins of which \(30 \%\) are 50 p coins, \(20 \%\) are 10 p coins and the rest are 2 p coins.
- Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of this population of coins.
A random sample of 2 coins is drawn from the bag one after the other.
- List all possible samples that could be drawn.
- Find the sampling distribution of \(\bar { X }\), the mean of the coins drawn.
- Find \(\mathrm { P } ( 2 \leq \bar { X } < 7 )\).
- Use the sampling distribution of \(\bar { X }\) to verify \(\mathrm { E } ( \bar { X } ) = \mu\) and \(\operatorname { Var } ( \bar { X } ) = \frac { 1 } { 2 } \sigma ^ { 2 }\).
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