3. The random variable \(X\) is such that
$$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$
Find expressions in terms of \(a\) and \(b\) for
- \(\mathrm { E } ( 2 X + 3 )\),
- \(\quad \operatorname { Var } ( 2 X + 3 )\),
- \(\mathrm { E } \left( X ^ { 2 } \right)\).
- Show that
$$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$