Derive or identify E(aX+b) or Var(aX+b) formulas

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

1. \(\mathrm { E } ( 2 X + 3 )\),
2. \(\quad \operatorname { Var } ( 2 X + 3 )\),
3. \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$

6 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } 3 x ^ { 2 } & 0 < x \leqslant 1
0 & \text { otherwise } \end{array} \right.$$

1. Determine:
   1. \(\mathrm { E } \left( \frac { 1 } { X } \right)\);
      (3 marks)
   2. \(\operatorname { Var } \left( \frac { 1 } { X } \right)\).
2. Hence, or otherwise, find the mean and the variance of \(\left( \frac { 5 + 2 X } { X } \right)\).

2 The random variable \(X\) has variance \(\operatorname { Var } ( X )\) Which of the following expressions is equal to \(\operatorname { Var } ( a X + b )\), where \(a\) and \(b\) are non-zero constants? Circle your answer.
[0pt] [1 mark]
\(a \operatorname { Var } ( X )\)
\(a \operatorname { Var } ( X ) + b\)
\(a ^ { 2 } \operatorname { Var } ( X )\)
\(a ^ { 2 } \operatorname { Var } ( X ) + b\)